Yeah sorry I'm in the same boat, I've only done the 80 and 90 BG's a handful of times. I would make the assumption though that it will mentor a 38 down to level 30 (as how the 80 versions mentors anyone 81 and over down). I do have a few pre-80 alts but I haven't taken them into a BG yet.
I guess the best advice I can give you would be ask in your servers level channels (level 1-39 in your case) and see. If you get no answer then you will have to try it and see what happens. Let us know.
As for being stronger...marginally yes. The same way that any higher level mentored will be stronger than a player originally at that level. However, most of your abilities/armor/etc have a checkbox in the top right corner that when checked will show you how the ability changes from PVE to PVP (less damage, shorter duration, etc).
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Things I sometimes play...
"What do you want to be when you grow up?"
"I want to be a unicorn!"
"Awww, why's that?"........
"So I can stab people with my face."