"And yet they hate us."
Translation:A přesto nás nenávidí.
Přesto is "yet-in spite of something"
Přece has many different meanings. Most often "nevertheless, after all, anyway as well as yet" Here it sounds weird. Not sure what one would try to say using it. The most famous use is Galileo Galilei"s "And yet, it moves" = A přece se točí.
It's the word order. The pronoun nás should go after the conjunction přesto. Your sentence is not strictly wrong, but it stresses "nás" in a way the English sentence does not. It's tricky, because you can find an excuse for using almost any word order in Czech if you push hard enough, but the stress and sometimes even the meaning varies a lot depending on the word order.