Tira's enhance simply gives an attack Multiple:1. Previous rulings have said that the attack gains additional instances of a keyword, so if I used Tira's E on Spinning Demon and had two momentum, over 2 enhances I could create 2 copies.

My question is simply, assuming I used the relevant E on the initial attack, would Tira's Multiple copy also gain the relevant ability/text? And, does the timing matter? For example, if I used Tira's E first on Blazing Fist, then the Combo E, would the Tira-Multiple have the Punch keyword?

All of the multiples get the ability when you play the enhance. If you do the enhance before the multiples the later multiples won't get it.

Correction to Aslum's answer:

The E on Spinning Demon creates two effects. The first is an effect that floats until it deals damage. At which point you pick up a character card if it does.

The second is a continuous ability that adds text to multiple versions of that card. That continuous effect lasts until the final multiple created from Spinning Demon is resolved.

This will be clarified in the next edition of the AGR.