RDM Verflare tooltip doesn't say what activates it.

[Iscah]

Once you have Verholy and Verflare up, can you execute both of them, or does using one clear the other?

You choose which one to use, according to your mana gauge. Using the one with the lower gauge is guaranteed to trigger Verstone/fire (whichever corresponds) while the higher one only has a chance of triggering it.