Be friends with or have a relationship with someone of a different political persuasion? Particularly, could you be comfortable with an extreme rightwinger or extreme leftwinger?
Different political persuasion, yes. Extreme views, no absolutely not.
Date someone from a different religion? Particularly if they are Hindu, or Muslim, or Jewish if you are Christian.
Depends how fervent they are regarding religion.
Could you be friends with or get along with someone who advocates for the death penalty (or adversely, is against the death penalty),
Advocates for the death penalty, absolutely not.
Is against or for transgender issues that differ from your views?
As long as they're not always banging on about it, yes.
Finally, could you be comfortable knowing, talking to, spending time with, anyone who has committed a serious crime in their lifetime even if they have done serving their sentence?
Depends on the crime but yes