Crate at Night- Door Open or Closed?

First off, he's too young to wait thru the night.  If I were telling you how to do it, I'd tell you not to keep him in the kitchen.  Keep him in your bedroom (all the time -- not just at night -- it's more conducive to sleep and rest cos the 'pack' sleeps there).  If your carpet is a concern, put a scrap piece of linoleum down under the crate and absorbent toweling over that.
 
But if the pup is in the crate then YOU will hear it during the night so you can get up and let it out.  See you really do NEED to let the dog out.  You don't want the dog to only have the option of going IN the crate.  So honestly?  Your days of sleeping 8-9 hours are o-v-e-r for probably a year or so. 
 
Yep, I'm serious and not 'nasty'.  That's just part of having a pup.  You won't get much sleep at first -- but that's part of teaching this dog to be part of the family. 
 
I usually tell people to put a bell on the dog's collar -- like a small cat bell.  Then during the night when the pup is rattling around, you will hear it and can get up and take it out (ON leash -- stay in control). You want to take the dog to that spot you've 'chosen' and only wait a couple of minutes.  If you let the dog out 'loose' you may never get back to bed!!
 
For during the day -- this won't be convenient or easy -- but you should consider having someone come in and let the dog out at least by mid-day (a 12 week old pup can't really go longer than 4 hours without elminating and that's likely a 'max' time).  But you'll make it less confusing for the dog and house-training will be far more effective if the dog understands quickly that it just plain shouldn't EVER go **in** the house at all.
 
But as far as your topic question "Crate at night -- Door Open or Closed?" -- my answer would be closed.  The whole 'point' of crating is to encourage the dog to hold it UNTIL you can let them out --- OUTside.  Beyond that it's pointless. 
 
You don't want the pup to decide to snack on carpet fibers, the corner of a blanket, shoes, or anything else during the night -- such leads to an obstruction.  You want the pup to learn to sleep. THRU the night hopefully.  It will happen eventually. 
 
A puppy left in the kitchen learns many things:
 
A. -- It's ok to poop and pee HERE, I do it all the time.
B. -- I'm lonely -- so I'm gonna bark!!
C. -- I'm BORED -- so I'm gonna dig on the cupboards.
D. -- Hmmmm I like the cupboard -- this piece that broke off -- I'll CHEW on it a while and get more off!
 
Even if you can get a neighbor or friend to come in, or maybe just a  professional (and bonded) dog walker - you won't need to do it forever -- just until you can lengthen the time alone without accidents.
 
But you can't have one set of rules in the evening about wanting the dog not to potty inside, and another for the day.  Therein lies confusion ... and a dog that won't be reliably housetrained.
 
This dog isn't old enough to make any 'good' choices for itself yet.  It shouldn't be chosing where to sleep, where to elminate or when.