I think what I'll do in this thread is deal with your first contention with the lottery first, and then we can come back to your second contention later once we've settled the first. That'll give me some time to mull over your second contention.
Sure! Sounds great to me!
By the way, I was pressed for time and so sort of skipped over the point you made about God approving the selection made by the cast of lots, not because He was controlling the lots but because it was random and therefore fair to all involved.
I see no biblical support for that idea. Surely you don't believe the the selection of Saul as Israel first king was actually random and how many lots were cast in a row with a negative result during the selection of David as Saul's replacement. It is seems totally clear that God was doing the selection and that the casting of lots was merely His vehicle for doing so that no only got the job done but gave Israel a means to express their faith in that fact. In other words, God could have just said in a booming voice, "David, the youngest son of Jesse will be my king over Israel!" and saved everyone a lot of time and effort but, as I know you are already aware, God doesn't like the results when He so overtly obvious like that. Things tend to go sideways when He does things right in plain sight like that even more so than they already do.
So, no, I think I'd have to reject the notion that God likes the totally random selection of leaders.
Lastly, one final tidbit to add to the argument for minimum qualifications needed in the selection process...
When the Apostles where selecting a replacement for Judas, did they make a completely blind random selection from the entire population of Israel or did they weed out people who would have been wholly unqualified for the position and then select randomly from a small number of well qualified men?