Guild Wars 2

Does Teq/Shatterer also have a Daily guarantee chest for each of them like Claw does?
Also does that chest have a exotic chance which leads to possible Precursor?

Yes and Yes.
All daily chests are replacement for the garanteed rare (or above) item in the normal chests. There is an ultra rare change you get a precursor.

Actually every thing you open/kill has a chance to drop a precursor, I had a guildie get a precursor from Shelter Gate(before the nerf, before the slight buff, before the nerf, before DR)
and a few guildies got it from dungeon chests.
in theory you could get a precursor from lvl 2 creatures in Queensdale (got lvl 80 rares from them a few times)

but that chance is even lower.