Don't want to give up on our Beagle but...

Wow...Can't give you any professional advice, but you've got a very bad situation.  Poor Abby seems to be in jail, and your kids are in danger when she's on parole.  Personally, I would consider rehoming the dog, as this situation is unfair to the dog and very dangerous to your children.
Good luck!!

Just so you know, if you take a dog like that to the Humane Society, she's VERY likely to be put to sleep.

The best thing to do for her is probably to get in touch with an experienced behaviorist to work with you and her, and work past this.

Yeah that's what I fear.  She's a nice dog.  Her cage is in the kitchen, and if I take her out but stay in the kitchen with here, she is very calm and nice, sits on my lap, and let's me pet her.  But take her out of that room and it's an entirely different Abby.  When me or my wife takes her for walks, she's doing really well with that now, almost walks perfectly with my wife.

Hi there []

It's great that you've managed to nurse the pup back to health. This is the reason that many of us on the forum refuse to go to stores that sell puppies - no responsible breeder would shove their 5-8 week old pups into a store to be put on display for their first few crucial weeks of life. But I'm sure you've found that out the hard way.

I do hope you decide to keep her, I'm sure she can be turned around.
How much exercise does she get on a daily basis? What kind of exercise? (dog park/on-leash runs or walks/backyard romps? All of the above? None of the above?)

And could you describe the "attacks" on your children? Does Abby break the skin? What do you do when she does this, other than crating her? Does she show aggressive behavior towards anyone else?

Knowing a few more details would help us figure out exactly where the problem is and hopefully get you on the right track to a well behaved pup. But as stated above, an experienced behaviorist would be a good idea.

I'm home most of the day with Abby, but so are the kids so no dog parks, it's closed whenever we'd be able to take here.  We walk her at night and whenever we can on the weekends.  We already know she needs more activity but we can't do it enough.  No, she doesn't break the skin but goes for their faces, scares them, knocks them down to the ground.  We try to give her time outs in another room and then bring her back, but after a few times she goes back in the crate.  No aggressive behaviour beyond that, except when anyone outside our immediate family enters the kitchen, then she goes off barking excessively, which thankfully is the only time she barks, not too bad for a Beagle!

Yes, no kidding Jennie but our children's safety comes first.  We've had training class with other dogs as well as private

Actually, that doesn't sound like an aggressive puppy at all. Just an understimulated, excited puppy. She doesn't sound like she's attacking your kids, so much as eliciting play. 

I'm not talking about training classes, and I don't think your kids are in any danger. A trainer is NOT a behaviorist. A behaviorist is a specialist who's been through years of schooling, and often has a doctorate in applied animal behavior. They're very different from regular trainers, in that they deal with mostly special cases, not just basic manners training.

Dan,
Is she food motivated? If she is, you may consider getting her a buster cube or a tricky treats ball. It will keep her mind busy for a while, which may be a good start. If she can figure out how to work one of these toys, it may help open her up to more learning.

Buster Cube: http://www.bustercube.com/

Tricky Treats:http://www.omegapaw.com/TrickyTreatBall.html

Both are pretty much the same idea. Dog plays with the toy, and the toy spits out treats. It might be worth a try.
I got Sierra the tricky treats ball, and she just loves it. I usually put her supper inside it, and then she spends the next hour or so getting it all out (though I do have to refill it a time or two in between). It keeps her busy  mentally, so I can be doing other things.

My friend has 3 year old beagle with same behavior problems. They use muzzle to calm him down (not metal or plastic - fabric with velcro).
They got it for walks, because dog picks everything from the ground. But sometimes they use it at home when beagle is out of control.
And it works perfect! He is down on his place with very sad eyes[]

Please don't give up on your beagle.

I can tell you I have been there. And having two young human children also
is a lot to hande.

Our beagle Buster was originally a pet store purchase surrendered to the HS
at about 8 weeks. He was a tiny thing. We kept him in the master bathroom using 
a kiddy fence and that worked well for him until about 10 months old. And then he
started digging holes thru the sheet rock in his spare time. One hole in 3 different
walls.

That began the crate training. Beagles are very food oriented. So, just like sit & stay, he learned the command "go inside" and he would go inside his crate for a
high value treat.

I remember those early times, & Buster too, seemed to be so energetic that
really any moment that I could not watch his every move, he needed to be crated.  

Just as you would teach your two children manners & what behavior is not acceptable, your beagle is just like them. When Buster was about a year old, he
would talk back to us when being scolded. By this I mean that if you looked at
him using a strong deep voice, he would bark and grumble back at us, like if he had to get his 2 cents out. Reminded me of my teenagers.

Abby doesn't focus in part because the beagle nose will take over their thoughts.

Buster at the age of 2 started to settle down. Being crated during the day, meant that he was no longer was crated at night. He started sleeping next to the bed, but
would manage to squirm his way into ours. This oddly enough, seemed to help
him learn to like comfort items without destroying them. Soon after a blanket
stayed in his crate without being shredded.

Buster is now 3 yrs. old. He does have couch potato tendencies. In the last
year, we have gotten Coco, to keep him company and someone to play with.
Buster will still love to find an open bedroom door so that he can rumage thru
the waste basket or find an old sock hanging around. He is still a thief.
Then he prances around the house showing off his lastest prized possession
that he shouldn't have.

In general, beagles tend to be good with children. I have cousins and neighbor
children in my house and Buster is no problem. Actually he would rather sniff
their shoes.
Possibly Abby is just not showing respect towards the kids ? Your 3 yr. old
could tell her to sit, etc. (As I would tell my son, to use his "Big Boy Voice")
when asking Buster to do something. Your son gives the command, but you
stand next to him and give the treat.
On the other hand, I would crate her while the children are eating, just in
case a food stealing incident doesn't turn into any other issues at least until
she settles down.

I guess what I am trying to tell you is that it Does get better, just it will take
time for Abby to mature. Like Buster, she still might have some goofy
behaviors, but after all it's a beagle.

To be perfectly honest, you made all the classic mistakes (but, that's how we learn).  Except in dog savvy families, it is often not adviseable to get puppies for children under 10 years of age.  They often do things that make pups fearful.  I think others have already mentioned why no one should buy from pet stores.  And, puppies need socialization and training during the critical period from 9-16 weeks of age.  The socialization part was not something you could do because of the pup's illness, but it almost certainly results in a pup that is not comfy with other dogs or strangers.  The training part could have helped, and I would have suggested that you engaged a trainer for in-home lessons.  That would have probably kept you from confining a young, energetic puppy to a crate - we prefer to use these as artificial dens, not jail cells.  If I were you, I would take the suggestion about calling a behaviorist.  If you can find a good trainer in your area to make a referral, all the better.  If not, go here: [linkhttp://www.ccpdt.com]www.ccpdt.com[/link] and see if a trainer near you can do so.
Your main problem is that a dog that gets little exercise (Beagles were meant to chase rabbits all day) is going to be nippier and jumpier than a dog that plays with other puppies.
If your dog is social with other dogs, day care is one option, or even "play dates" with someone from your training class.
You could try to establish a connection with the local Beagle rescue in your area - these will be people who are knowledgeable about the breed and can help you find the resources you need to be able to keep your dog.  In the sad possibility that you can't, maybe they can help rehome him to a Beagle-friendly home.  Unfortunately, this is a breed that is often used by medical labs, and there are way too many of them (which is why I suggested opening a dialog with the rescue - to solidify your position that you are really trying to keep this dog, and will only reluctantly give him up). 
This is another sad tale of a pet shop dog - I hope you will tell anyone who will listen NOT to go down the same road.  It really sucks to learn this by experience. [sm=sad.gif]

This scenario is one of the reasons I'm an anti-crate person. I've seen it over and over again-- puppy is out of control, so puppy gets locked up in crate. Pup gets wilder and wilder whenever let out, so pup gets crated more and more often. It's a horrible cycle. Usually the dog ends up spending 23 hours a day in a cage, and then eventually gets shuttled off to a shelter. A pup in a crate learns nothing.
 
You need to break this cycle. It will be a difficult transition. I would suggest in future only crating the pup at night or when no one is home, and using the "tether" technique instead-- basically the pup is on leash tethered to you or to some object near you at all times, so he can't get in trouble. But he's out and about and learning. Carry his food or some treats with you, and while you are tethering him about on your daily chores whenever you notice him doing something good (like lying quietly, not chewing the furniture, not mauling the kids: praise and treat).  Before you start the tethering though, hire a babysitter for the kids, and devote half a day to "tiring out the dog" every day for a week. Go to the dog park, take the dog running, take the dog to the petstore to walk around, anything you can think of. Bring the dog home, begin your tethering training. He'll be exhausted that first week and hopefully won't give you any trouble, and you'll be able to reach his mind and teach him some manners that will hopefully carry over to the next week.

I think others on this board will give much better advice than I ever could, so I'll leave that to them.
 
Though, I do want to say that though what these wonderful intelligent people have to say may seem overwhelming and threatening..they don't mean it like that at all.  We here love dogs veyr much and want to see the best for them.  Many of us woudl love to see you work things out with yoru dog for you and your children's sake as well.  I know that I would love to help give others the chance to have a wonderful relationship with their dogs as I do with mine.  I really do hope you try to take some of these peoples advice and work things out, as IMO your dog simply is untrained and easily excitable.  And with proper training, it isn't impossible to correct.
 
I wish you the best of luck.  []