Have you tried rushing your W first? I hadn't done that until recently and it really makes a world of difference for defense at low levels. Instead of the shield breaking in a single auto-attack it ends up holding off a few attacks and providing very meaningful armor and magic resist. I don't rush W in the jungle, but I figure it's an option you can try.
I have a question regarding the timing and of when to use his abilities in his jungle - should I spam QWE everytime it pops off, or should I let the 2 passive effects wear off before using next ability, 2 AS, another ability, 2 AS etc?
"Life is like a bowl of soup. You only get blown if you are hot."
Lee's a champ you need to play and watch to learn and get good with. His kit allows for so much variation and has such a high skillcap in comparison to any other champion in the game (Link's famous Lee Sin escape is what comes to mind first of all) that having someone tell you is not as helpful compared to say someone more simple like Morgana. You can still learn a few basic things though.
Watch players like Froggen or Chauster stream, their Lee's are some of the best in the game. You can pick up a lot of things from watching them.
Blackroo was pretty spot on. You can top, jungle, or mid him (First two are much more common). If you're Jungling Lee, you're going to be taking a supportish role due to lack of gold. Wriggles, Phage, and an Aegis of the Legion are all good pick ups that give you tankiness, some damage, and the sticking power to protect your carry and deal some damage.
If you play him top, you have the option of going straight damage (or stick with a tanky/supportish role, whichever you want). A BT and LW will make you an absolute fucking terror. Grab a GA for additional tankiness and your an incredibly difficult to pin down target that has some damn scary assassination capabilities.
Also carry wards for 1337 jukes and moves. Nothing like safeguarding to a recently placed down ward for the slow and kill.
Continuing on that, there's no reason at all to start with Q if you'd want it for that reason. Take W, or E (I prefer W) first, and get Q for your lvl 2 gank. I honestly feel that W > E > Q is a better way to go early game.
When would you ever think it's better to max something other than E first? It's got an AoE slow, decent damage, and helps you clear faster.
Another reason for not maxing E except for less damage is that it drains your energy really fast if you are using it for the damage.