A lot of people say there is a "server tick", but in my experience with most mechanics, avoiding an aoe by the time the cast is complete means you avoided it. Sometimes you have to avoid it earlier than that the higher your ping is, but that's it.

There isn't a thing where you can avoid an aoe before the cast time and still get hit (unless it is one of the few mechanics in the entire game that works differently). Just never had that at all. For people that do, it's just ping or a computer/ISP that delays network traffic.

Weaponskills/spells are instant, but abilities are slow because there is an animation delay. Note very carefully the difference between a weaponskill/spell and an ability. An ability can go through a whole animation before finally executing the effect.

You probably experience that the effect of an ability doesn't display the result (deaths, stuns) immediately. I think part of that is the animation delays and part of it is ping. In pvp, by the time your stun reaches the server and communicates that it worked to you, the player has moved a long distance and then finally gets stunned. Maybe there is an actual server tick for the result of your actions, I don't know. But one way or another, this tick doesn't affect your ability to avoid attacks before a cast finishes.

I notice the effect of ping most in pvp, chocobo racing and the delay on pressing abilities.

There is a server tick for dots and they tick every 3 seconds.