Couple of questions about Diana

Buddy you just sang my song. I am a jungle main, and Diana is my favorite champion. TLDR: God yes. To both.

Jungle Diana is the most fun I have had with any champion, but it does take some skill and practice. The most successful build I have had, as well as the most consistent is Spectral wraith -> boots - Tooth - sorc shoes - Zhonya's. This gives you excellent burst, sustained damage, dueling potential, playmaking potential, and allows you to split push should your team need it. (Note: Jungle Diana can pubstomp the shit right out of a khazix in his own jungle if played right.)

Some tips: Starting shield gives you the best initial clear and keeps you the healthiest. Additionally while you want to max Q first, you want to get two points into your shield early. I HIGHLY recommend going W-Q-W-E-Q-R-Max Q. Not only does this make your ganks much more potent (and safer) this will keep you alive if you get caught, and allows you to win duels against other assassins and jungle bullies. IE this allowed me to kill an equal items/level Kha that got the jump on me while I was isolated, and come out at half health. Your pre six ganks will suck. FYI. Farm to six, then never stop ganking.

I think Nashor's is great for the cdr, but the autoattack passive might not be the most helpful depending on the rest of your build or the play style you're going for. If you want burst damage, you'd be better off building dfg and other raw damage items. Nashor's seems like it would be a good option if you want sustained damage. Core items in a build like that would be rylai's (or iceborn gauntlet) and liandry's. Also Nashor's would fit well if you plan on jungling with Diana (she can jungle but she's not the best).

1. if u want to go into a more tanky bruiserish direction, it'S great, if u want to go fullap-assasin, it's bad.
2. She's quite ok, probably profits rather much from the preseasons change. Her clear is good, the dueling is decent, but she'S manahungry. post 6 she becomes incredible dangerous.

1. Depends on your opponents. If they're squishy, yes. If they're tankier, you may want to bulk up yourself.
2. A bit harder to answer. She CAN jungle, much like Akali can jungle. The issue is that there is no reason not to take a better jungle. Unless the opponents are crazy overextended, her ganks are lackluster until level 6. And at level 6, there are other champions with much better ganking potential. Her clear is actually pretty good, but you'll have trouble without blue buff, and again, if clearing the jungle is what you want to do, there are still better junglers for that as well.

Nashor's Tooth is good on her, yes, but not ALWAYS. It offers good dueling potential, and the cdr is fantastic, but sometimes you need to be burstier or tankier, and in these cases, you'll want to take other items.

As for if she can jungle, absolutely, HOWEVER, you need to realize that her pre-6 ganks are mediocre at best (she can really only gank when the enemy is WAY overextended and she's free to use moonfall to keep them close) so you need a team that's willing to do without ganks very early on. Also to keep in mind is that she is extremely mana hungry (when full comboing frequently), and will typically need to have blue buff for most of the duration of the game.

So I guess I'm saying, she is not the best jungler, but when well played can carry a game, especially because of her versatility. She's able to be bursty, tanky, and even sustained-damagey if you alter her build to match what you want to be doing.

Zhonyas is ALWAYS mandatory, don't ever leave home without it!