Housebreaking Whippet pup - secret little pees

Ok, replying to my own thread...
I actually JUST caught him doing this.  Someone said he was marking his territory? 
He was sniffing around a strange area and then went to squat.  I said 'NO!' and the grabbed the leash and took him outside to pee and then lavished him with praise and a treat.  Is this all I do to nip this in the bud?

He's not really old enough to mark, yet.  He's being a puppy. 

If he was bought from a store or a byb, then you may have some issues with the training.  Paper training, IMO, is a waste of time and of effort.  It teaches the dog that sometimes it's ok to go in the house, and forces the pup to make the decision.

Housetraining 101 is what you need.  Restrict movement, and no unsupervised time at all.  Tether him to your belt on a 4 or 6 foot lead.  Anywhere you go, he's there.  If you can't be with him to supervise:  put him in a crate or X-pen.  Four days is not enough time to teach him what is "den" and what is not "den."  He's choosing areas he doesn't normally go into to eliminate waste.  That's because he doesn't know that those areas are also den.

A puppy this age should go outside after every meal, directly after being taken out of the crate, after play, and before bed.  Say goodbye to an uninterupted night's sleep.  He needs to go outside every 2-4 hours for the first month.  This is the best time to attach a command to the behavior.  "Hurry up" or "go potty" or "do your business."  Say these commands as he's making.  You'll have to use a seperate word for #2.  Dogs don't correlate ing and pooping as the same thing.

You have 10 months of hard work ahead of you raising this puppy.  What you get out of this pup depends on how much work you put into him.  Good luck and we want pics!!!

ETA:  The word "NO" doesn't tell your puppy what you want him to do.  Rather than grabbing the least, walk over calmly and pick him up to take him outside.  Redirection, rather than correction, at this stage give your puppy confidence in you and in himself which he will take into hood with him.

Xerxes has already given some great advice, I just wanted to add make sure you don't say "No!" unless you actually catch him in the act, and even then an "unh-unh" will often do- otherwise he'll learn to hide his pee (or do it when you're not looking).
Also if he's just peeing in certain spots, make sure to clean them properly with an enzymatic cleaner to get rid of all the smell.
 
Good luck with your puppy! I love whippets!

The whippet came paper trained.  In the pen at the breeder's they had a dog bed and paper and the pups only eliminated on paper, not in their dog bed.  I'd rather he have an accident on newspaper than my beige carpet if I have a choice.
You make a very good point about the fact that he has a choice of toilets though.  If he doesn't want to hold it he can simply go on the paper.  I've been home with him for 4 days so he hasn't gone on the paper much because I keep taking him out every couple hours. 
Should I slowly start to remove paper so that he has less and less paper to go on?  Should I slowly "phase out" the newspaper in the kitchen?  And if I do that gradually, who's to say he won't just go on the kitchen floor where the paper used to be? 
We don't crate Leroi.  He doesn't destroy the house and just loves to lie on pillows on the couch, like a typical whippet.  He sleeps on a fleece blanket on the floor beside our bed at night and we take him out overnight.  He can hold his bladder for 5.5 hours while sleeping with us which really impresses me. 
Our crate is a carrier/crate.  It has a tent-type zip door, not a cage door, so if I were to crate him to teach him to hold his bladder it would be like putting him in a black box, you know?  I go back to work tomorrow.  He will go out at 8:30am and I will be back again around noon to take him out, but I'm thinking I'll be coming home to accidents in the kitchen because the newspaper will be there.  Obviously I don't expect an 8.5 week puppy to not have any accidents already, but if he were snoozing with me between 8:30 and noon, I know he wouldn't go pee until I take him out.  When he's left alone I'm sure he'll just get bored and wander into the kitchen and take a leak. 
Any suggestions?

If you do not crate your pup, then restrict his movement with a baby gate or two.  Keep him inside the kitchen, do not give him free reign over the apartment.  That is a disaster waiting to happen. 
 
If you are already having "little accidents" then they will probably get worse.  Especially since he's being left alone.  You are giving him too many decisions to make at once, without a caretaker there to provide him with direction or re-direction.
 
Remember, just because he hasn't chewed the couch, the walls, the cabinets, electrical wires, the table, the carpeting, or anything else up to this point, does not mean he won't.  It just means he hasn't had a chance to "discover" what wonderful things they are.  Restrict his movement to one room for a few weeks.  If he does well, give him a bit more freedom. 
 
It's not an easy thing to do, coming from an emotional human viewpoint, but it is the best thing to do for the health, welfare and training of your whippet.

But we've already let him set up shop on the couch and into the bedroom on the floor with his blanket.  The first nite we baby gated him into the kitchen he SCREAMED for hours.  I live in a 1-bedroom apartment.  I didn't sleep a wink.  If I gate him he'll cry all night long.  The first night we got him I let him scream for an hour and a half and when I finally went to see him he was panting and his tongue was hanging out because he'd worked himself into such a mess.  When we let him out he slept quietly on the couch. 
I'm already at my wit's end what with all the changes to my life that getting a puppy has caused.  YES, I waited to make sure I wanted one, but right now I have that "want to send him back" feeling that I hate.  I go to sleep with my stomach in knots every night. 
Please just lay out what you think I need to do, I'm desperate here.
My questions:
Should I remove the newspaper immediately?
Is it ok if I can't close his crate door?
Should I baby gate him in the kitchen overnight even though he screams loudly and I can't sleep? 
Should I baby gate him when I'm at work? 
To eliminate those mini-pees on the carpet I guess I should just monitor his every movement and say No and take him outside when I catch him squatting? 
(I use positive reinforcement when he pees/poos outside)
 

These are typical puppy problems, and are relatively straightforward to fix, if you are consistent. Try not to stress out about it too much, and make sure you enjoy the puppiness whiile you can . Sometimes it can be really tough, but thats why puppies are so cute (so we don't send them back)[].
 
First of all, I would definitely recommend babygating him while you're at work. My dog's fine with full-run of the house at night, but if I leave her out during the day then she goes into "destructo-mode". Also, if he still hasn't got house training down, babygating will help with the messes.
 
If Leroi's still using the newspaper fairly reliably, then I would start gradually moving it towards the door, tho I don't really think there's any harm in taking it away completely if you supervise him all the time, and crate him when you can't (as well as taking him out every few hours). In this case, then you would want a crate that you can shut.

I totally understand about the screaming/barking/crying.  I raised a pharaoh hound from pup to adult.  Think loud, turn it up a few notches then you're close to what it was like. []
 

Should I remove the newspaper immediately?
Is it ok if I can't close his crate door?
Should I baby gate him in the kitchen overnight even though he screams loudly and I can't sleep?
Should I baby gate him when I'm at work?
To eliminate those mini-pees on the carpet I guess I should just monitor his every movement and say No and take him outside when I catch him squatting?
(I use positive reinforcement when he pees/poos outside

 
I don't think you should remove the newspaper immediately.  If he's used to it and he doesn't find it when he needs to go, you probably won't like the results.
 
When you say you can't close the crate door, can you secure him inside?
 
Overnights, to let you get some sleep, and keep your neighbors from killing you, I would either move his crate next to your bed, or leash him to the bed.  At this age it's not the best thing for him to be sleeping next to you on the bed-due to problems that might creep up later in life.
 
While at work, he should be babygated with access to his bed, a toy or two, and depending upon the length of time-minimal amounts of food and water.
 
To eliminate the extra-curricular peeing, keep watch over him as much as possible, keep him close to you.  I would hesitate with the use of "NO" as much as possible.  When you say "NO" during a time when you catch him peeing, that same "NO" won't translate to "don't jump up" or "stop barking" or "stop chewing on my shoes."  So try a "No pee" or as another poster suggested "unh uhn." 
 
You'll find out how sensitive sighthounds are to correction as he grows older.  They don't operate well under negative conditions.  Also, please find a positive reinforcement trainer for basic puppy class and obedience.  Your dog will benefit greatly from it, so will you.[]

Remember too that you have only had him four days at this posting, and that you are setting him up for a whole puppyhood. What he is comfortable and safe with now, as an unsure newcomer in the house, he won't be in the future--trust me! So it's important to set up a protocol that meets your needs for the next year... and stick with it!