Can avatars invoke soulshards to affect the die rolls when generating a creature?

No.  Well... officially, that is.  Let's be honest: you're going to change the rules however you want to make the game more fun.  So go with fun.  If it makes you happy, then go for it!

How do I invoke a soulshard?

You simply declare which soulshard you are invoking and for what purpose, then pay the soushard to the common bank.

What's the deal with soulshards and masteries?

A mastery has two parts: a main effect that is usable by the avatar at no soulshard cost, and a secondary effect that is activated by invoking a soulshard.

A soulshard may be invoked to activate a secondary mastery ability at any time.  During combat , there are restrictions to how many soulshards an avatar may invoke each combat phase.  These restrictions apply to the number of soulshards invoked, not how they are invoked.

Can my avatar invoke a soulshard and drink a potion in the same combat phase?

Yes.  Invoking soulshards does not require an action.  The same goes for using a scroll.

How many soulshards does it take to get to the tootsie roll center of a tootsie pop?