Up to 1000 (without artifact), anything that shows Powerful Reward or Pinnacle Reward will raise your level. After 1000 (without artifact), only activities showing Pinnacle Rewards will help you until you reach 1010 which is max right now. The Clan reward from Hawthorne is actually a +1 Pinnacle reward too. Also the Raid if you are up for it.
No problem. If you’re working on more than one character make sure to move your highest guns to your 2nd/3rd characters “before” you begin playing with them. Always have your highest guns on the character your using. They don’t have to be equipped, just on the character. That raises your average and your drops will be higher, unless your drop happens to be another gun, which sucks.