Does the level up occur before or after clicking the level up button?

Do we level up in Skyrim only after clicking the level up button?

As Ben said, you can still gain experience and level up individual skills. However, yes, your level only goes up after you have selected to level up. The wiki offers much more information.

To answer your other question:

Will the enemies stay at the same level?

For all intents and purposes, yes. There is a really good answer already for this here, so I will not rehash.

Yes and no.

You still do gain experience, even if it doesn't change the XP bar. However, until you choose your stat to improve, you still are effectively the same "level".

Your skills however, do still level up, allowing you to apply skill points to other level skills. Again, however, this is only accessible after you "level up".

So, if you choose not to level up, but run around and kill enough people to gain an additional level, you will level up twice.