I've never sold anything on eBay, but I've done a fair amount of buying and "policing" (with Dale)...
1. I wouldn't say it's "normal", but it definitely happens from time to time when buyers contact you for a "side deal". And I do
think it's against eBay's rules, if it works out where eBay gets removed from the deal and isn't going to get their cut. They definitely want their slice of the pie!
2. It's unfortunately normal that there's a lot of idiots and jerks in the world who'll throw a hissy fit, and lob various threats over the net if/when they don't get their way with something. I say just report them to eBay, saying that they're soliciting the sellers and trying to cut side deals without eBay's involvement. They'll be hearing from eBay in no time!
3. All the time? No. Every now and then? Unfortunately, yes.
4. Either they're just getting nervous and upping their bid to stay ahead of any potential future competition, OR they're "chasing a reserve" amount, if there's a reserve price on the auction. Buyers will often up their bid little by little to try to reach the reserve amount without going too far over it, thus making the auction "good" to assure themselves that someone's going to collect the loot!
Good luck Scott!