Level sync system

[Rongway]

How does it actually work in terms of stats and equipment? I can see that my level obviously syncs and above-level skills become unavailable, but what if I had top-tier armor and weapon? How does that syncs down?

Gear over the item level sync for the dungeon is syncked to the item level sync for the dungeon. If the dungeon doesn't have an explicit item level sync, the item level sync is the max item level available at the character level you are syncked to. Materia on item level syncked gear is ignored.

Let's say I, level 50 Paladin, try some 20-level dungeon. I have a 50-lvl sword and a 30-lvl sword. Would wielding a 30-lvl sword make any difference compared to 50-lvl sword, since it's a 20-lvl dungeon anyway? Same goes for armor, of course, and stats they give to character.

Both swords will IL sync to the max item level available at character level 20-something and materia will be ignored. The only difference will be in which secondary stats are present. If you had a third sword that was the maximum IL available at the dungeon's sync level, you could equip that sword and it would not IL sync, and its materia would be active.

The most powerful gear in an item level sync context will be gear that is exactly the IL of the item level sync, with full materia melds. This is why people working on Ultimate fights keep a set of fight-specific item level gear for each fight.

And I guess same system works for FATE events, or are there any differences?

There's no difference. FATEs use a character level sync with no explicit item level sync, so the item level sync is implicitly the highest available at the syncked character level.