If you have solid guns and use the right elements, you should be able to tear through things, especially in normal mode. Since you're playing Zer0, you might want to change your playstyle up a bit, given his fairly niche skill trees. If you can push through all the way to Many Must Fall, you'll be golden. If you're having trouble with constructors, you might want to spec into b0re, and practice hitting their weak spot with it. Also, a note, since you mentioned getting instantly killed. You can't get instakilled in this game. It's literally impossible so long as your health is above 50%. If you have access to a Moxxi weapon, any healing skills, or transfusion grenades, use them to make sure your health stays above 50%. Also, if Zer0 [i]really[/i] isn't working for you, you could always just start over and try a different character. Salvador is easy-mode, and Maya has great mobbing skills from the get go.
Nope. I die instantly to things. My health goes from full to empty in 1-2 hits. Or it's a bunch of hits it succession that melt my health from full to empty in a couple of seconds. It just happened while fighting a bunch of threshers. I went from full health and shields to dead in 2 seconds.