How many sentences are correct?

1. Correct.
2. I guess it's grammatically correct but it sounds as if someone witnessed the burglar's whole escape. 1. would be strongly preferred.
3. Correct. Similar to "I saw him walk into the room".
4. If you mean that you noticed him when he already was in the room, I would use "had walked" to put emphasis on the sequence of events.

I find Number 3 unnatural.

I hesitated for a while over 3 to be honest, but if you can say "I saw him walk into the room" or "I watched him walk into the room", then I can't put my finger on why "I noticed him walk into the room" would not be acceptable. All of them are verbs of perception, but now it seems to me that "notice someone do something" is just not so common.

ozzourtiI hesitated for a while over 3 to be honest, but if you can say "I saw him walk into the room" or "I watched him walk into the room", then I can't put my finger on why "I noticed him walk into the room" would not be acceptable. All of them are verbs of perception, but now it seems to me that "notice someone do something" is just not so common.

My probl

enoonMy problem with it is just that it sounds wrong to me.

Well, it may not be very common but I am loath to call it 'wrong'.

What do others say?

I'd say that when Carly Simon sang " You walked into the party like you were walking onto a yacht" she definitely noticed him walk into the room.

Some people "make an entrance" and when they do, they want people to notice them walking into the room.

There is nothing wrong with that sentence at all, in a context where the entrance was a "noticeable" event.

BarbaraPAthey want people to notice them walking into the room.

Right. "Walking", not "walk". That's the very point at issue.