FASA Games, Inc.

When a wizard uses Flame Flash, it says that the "Effect test determines how much damage is inflicted each round" and that the standard duration is 2 rounds. This is how long the flame burns for, right? In other words the Wizard casts the spell, lights up their target and then next round they are still burning. I'm deducing this is the case and that the wizard doesn't need to stand there shooting flames for another round because there is no mention of concentration or of changing targets, but I thought I'd ask since it didn't seem to be clearly stated.

Your initial assumption is correct. Target takes damage twice, first time on the round it's hit and again on the following round.

Yes, no Concentration needed, so can cast other spell in second round. And you roll each round Effect Test.