Advice needed: Relic +1

Why do we do it before getting AF parts? Could someone enlighten me why this is the norm for most?

Because the increase in weapon damage is going to be a bigger improvement than the increase in stats on the armor. It would be wisest to do it in this order, but not 100% necessary.

And because allagan weapon drop in T5 of COIL only

But the difference between thyrus and thyrus+ seems pretty small. Why do I need to worry about damage if I'm healing?

because weapon damage affect how much heal you can do and
3 damage = 21 mind (approx)

It's incredibly small, it does not matter unless you are really into pure efficiency.

If you roll into Coil, as a healer, with Relic+DL - you can beat Turn 4. I'm less sure about T5 since I'm fairly inexperienced there.

Appreciated guys.

First three items to buy with your myth are weapon +1 because it is the highest upgrade to your healing of the options you have available. Next are pants because they drop in Turn 5 and are a large upgrade in health etc. And then ring because when combined with the ring from turn 4 should be your BIS ring set. After that its kind of a crap shoot for which piece to get because you run the risk of getting a allagan piece that overrides your myth piece. You can maybe jam a head or chest piece in there if you get an allagan piece as well since you are most likely using the DL cowl and will need an accompanying piece.