If I will heat the stone in the oven, will it retain heat long sufficiently once out of the oven to make a few crepes? Will the dough not stick to the stone and will it distribute uniformly or it will form patches?
I think it's a bad idea...
Crepes are made with a batter (as opposed to a dough) spread thin over a hot metal plate (seasoned or oiled).
A baking stone has a porous surface and I suppose the batter would just get stuck to your stone. It doesn't happen with a dough because it has enough structure to not fill every pore of the stone on contact.
On the other hand, if you had a baking steel instead... that might work, they usually have enough mass to hold heat for some time (and they don't crack).
(disclaimer: I'm not associated with the brand, but I do have one of these and it's awesome)
Pre-heat it, brush oil over it and spread your batter. I'm not sure how many crepes you can do before having to send the steel back to the oven, and then it will take a while to have it hot again, so it might not be practical.
No, don't do it.
Good crepes are made within narrow parameters of heat exchange. You can observe this when making crepes on the stovetop - the first crepe is almost always bad. The pan seems to be either not hot enough, or too hot. After the first one, it somehow "stabilizes", or extra heat starts to creep on you. In the second case, it will get too hot after a while, and your crepes will start throwing bubbles on the first side and burning on the second.
If you are in the good zone though, and decide to reduce the heat - say because you have been making nice crepes for the past 20 minutes and now it is your last crepe and you want to turn off the plate early - you will end up with a bad crepe again, even if you think that you have enough heat left over from keeping the pan (and the plate below it) hot for so long.
Given that crepes are so sensitive to temperature, I wouldn't even start experimenting with your stone setup. If you don't have a stove, get a portable plate, they are cheap and versatile and don't take up much space. I used to have an induction unit that was quite nice and made good crepes.
All of the baking stones I know need to be slowly heated and cooled down - i.e. they go into the oven before it's turned on, and stay in the oven (with the door closed) after it's turned off. Otherwise you might find yourself holding pieces of your baking stone, with the rest of the heavy and hot stone falling on your feet, wooden floor, pet, etc.
So don't try this for safety reasons.