Are Lava Mobs Grinders Allowed

Yes.

 

This is a good point although i belive you only get exp from killing the mobs with a sword. I am not sure of this but i want to say that is the reason

 

You only get xp from sword killing so lava farming and getting the drops won't make you much money

 

Yes

 

If they are killed without interaction, do you get anything from it?
xp isn't dropped from non-player kills, so that is not counting.
IF you create a mechanic that allows you to somehow retrieve the items, it's illegal - you are basically generating money without any work. Spawners in which you need to one-hit the mobs are legal though.