If I have a normal version of a game (in this case Rome : Total War/ Rome : Total War Barbarian Invasion) and I want a DLC (Rome : Total War Alexander) from Steam, will it work with my other games that are not from Steam?
No, you cannot purchase DLC on Steam when you do not own the Steam version of the Game. For some older Games Steam does accept the CD Key to unlock this Game on Steam though.
This is not the case for the original version of Total War : Rome though. Only Rome II will be accepted.
I'm afraid the DLC won't work with the normal version. This Steam forum thread and also this one suggest that it is not possible to buy DLC for non-Steam games. There are a number of retail games that don't require steam, but can be registered and added to your account with codes that can be found here.
Unfortunately, Rome: Total War is not on that list and it doesn't seem to be enough to just add it as a non-Steam game into your Steam library to download DLC for it.
No. The Steam store page clearly states (emphasis mine):
Notice: The Rome: Total War Alexander Expansion requires the Steam version of Rome: Total War Gold to purchase.
If you add it to your cart, and try to purchase it, you will receive the following message:
Your transaction failed because you are trying to buy 'Rome: Total War™ - Alexander ' which requires ownership of 'Rome: Total War™'. Please correct the error and try again.