Persecutor

Can I brand an enemy and step toward the closest Evil on the same turn?

No, you must choose only one option each turn.

Can I brand an enemy in a different region, or one with Stealth?

Yes! Branding does not target, and therefore you can choose minions in different regions and/or those with Stealth. This can combo successfully with other cards that care about minions being Evil!

Can I brand an enemy who can't be modified, e.g. Monks of Kobalsa?

Yes, you can still brand them, and your cards will still treat them as Evil. They aren't actually being modified or becoming Evil; your cards just treat them that way.

If my opponent has a warded minion and I brand it as Evil, does it lose Ward?

No, because only your cards treat it as Evil. Also, branding does not target, so won't break for that reason either.