Hello Paperbell, and welcome to the forum.
Element choice only affects your starting ability and resistance.
A demon's modifier does not directly power up related moves, though most modifiers (especially non-unique ones) tend to increase stats that are usually important to that element's moves. Firesworn is a hybrid Slash/Fire modifier that behaves a bit oddly when it comes to stats: in essence, it tries to make your stats more "average" since that is beneficial for a Slash/Fire hybrid, which needs every stat. If Firesworn was used on a character with low Magic, it would raise Magic, making Throw Flame stronger. But, if it were used on a high Magic character, it would actually lower it and make it perform worse. The Fiery modifier on the other hand is a pure Fire modifier, it always improves Magic, and would always make Throw Flame stronger.
How you handle demon fusing / ability copying depends a bit on your relic and strategy. I will agree it's generally good to have a strong healer (or even two, so you can rotate them out if SP becomes an issue, or use both if the incoming damage gets crazy... though this will limit your own damage in turn!) unless your summoner is a healer, but I've never been as sold on the Faerie->Gandayah advice as some folks.
For example, a notable thing is that all three starter demons begin with the Draw Wounds ability: you might look into trying to incorporate this into healing strategies. Yes, it damages the user... but the free action aspect is extremely powerful. On any demon at all, it allows them (without using their turn) to give HP to those in need freely. On a demon with life-draining or other abilities that are self-healing (Reshape, Flesheater, Souleater, abilities that use Regen or Guilt status, etc.), it can give them a way to share that healing with others. On a demon already good at healing, it can allow them to heal two targets in one turn, or one target twice, in dire situations. (Refresh, the SP version of Draw Wounds, can work in similar ways... though in that case, since self-SP recovery is much rarer, you'll probably be looking to put it on demons that don't use much SP, such as some melee attackers, or heavily cooldown based demons.)
And yes, Evasion abilities stack.