I can only answer a few of your questions as I always keep autorepair on and haven't run into all those situations.
1. Where can I see if my equip has been damaged?
You can see it on the website, not sure if it's updated in-game after it breaks.
2. When I leave after the round started with wearing gear, does it damage?
It doesn't get damaged if you leave before the first XP/gold tick - that would be in the first minute.
4. Can I change my gear after I have died? Or after the round when the Equipment pops up? Or is my gear getting damaged then?
Even if you change your gear after death it doesn't matter, the equipment you spawned and died with is the one that gets the damage.
5. Was the sellprice lowered to 50% of the buying price? Or is my Equip damaged and I cant see anything that it is damaged?
Sell price was lowered to 50% of the buying price.