it’s too hard to finish PVP daily quests because not enough player in Battleborn PC 
I can’t wake up in NA or EU daytime and when I playing in NA/EU server, I have very high ping. damn my internet
I don’t think you can. You can in Bots Battle and in private PvE though, if the quest is applicable to PvE.
You can’t. Only in normal versus matches or Bots Battles.