Polish e is pronounced like 'eh', or like 'e' in the word "pet"

It's generally the other way around. Polish tends to devoice last letter in the word if it is a voiced consonant, so hypothetical "brad" would be read as "brat". However, you voice or devoice the whole consonant cluster spanning multiple words depending on the voiceness of the last consonant in this particular cluster, eg. " Brat z żoną" would be pronounced "Brad z żoną" (cluster "t z ż", ż is voiced); "Brat z siostrą" is pronounced "Brat s siostrą" (cluster "t z s", s is unvoiced); "Rycerz z tarczą" is "Rycesz s tarczą" (cluster "rz z t", t is unvoiced) and "Rycerz z giermkiem" is "Rycerz z giermkiem" (cluster "rz z g", g is voiced).

In English you can shorten "is" to "'s" only with personal pronoun, eg. "He is writing." -> "He's writing." Shortening it with a noun is colloquial and that could be the reason it is not allowed here.