Guild Wars 2

Will transferring servers have an effect on the chest at the end of the season for the achievement?

I’m guessing transferring before the season is over will have an effect, but I don’t know.

If I transfer after, will the special finisher we get with the chest change as well? For example if I get one for my server finishing first in bronze, but after the season is over I transfer to a silver or gold server, will I lose that that finisher?

No, You are linked to the first server you logged into when seasons started. If you started on Anvil Rock and after seasons started you transfer to Blackgate, you get Anvil Rock rewards.