So I have always wondered if its possible for Demi-Gods to exist in TES? Can gods have children with mortals or children at all?

Gods have to sacrifice actual parts of themselves to propogate. Most of them died doing this in creating Mundus. The remaining gods (at least the ones called Divines) are kind of the survivors of that mess, if you will. I doubt they're willing to try again.

What about an Avatar of a divine?

It's an interesting concept. One of my first characters was an eyelash of Sheogorath that was plucked and flung to Tamriel, and took the form of a Breton. Kinda Greek Mythology, but I like it.

Someone like http://www.uesp.net/wiki/Lore:Umaril_the_Unfeathered could possibly be considered a demi-god.
There are carying degrees of godhood and power in TES and the lines are blurry.
Some mortals like Mannimarco ascend to godhood but there are also very powerful 4000 year old wizards like Divayth Fyr

Hmm this is interesting. Can the dragon born be kind of considered a Demi-God? I know he has a soul of a dragon but isn't Akatosh the father of all dragons?

As Merari has said, the lines are blurry when it comes to godhood.

I would personally consider the following demi-gods of varying power
*ALMSIVI - They owed their power to Lorkhan's heart
*Umaril the Unfeathred
*The various Shezzarines
*Mannimarco maybe.

A case might be made for Mankar Cameron too (at least when in Oblivion).

Edit: Forgot Dagoth-Ur, for similar reasons to ALMSIVI.