Let me stop you in your tracks mate :)! The thing about eve is that she is squishy and is played poorly very often and is easy to mess up. People misunderstand her for good reasons because she rarely shines and ends up feeding.
I have had success with her jungling by how she comes out of stealth and has the slow along with her e which shreds armor. Her ult allows her to gain health after a kill and move super fast. That is a perfect jungler description: a champ who can gank well and have some way to slow to have the laning champion(s) catch up.
I have had some success laning by being aggressive early (know your limits however) and then farming up while they stay back, some teams have eve roaming nonstop and a jungler on the side, while having only 1 champ in each lane.
It is true though she isn't really played in tournaments, but I have seen her in the high elos (2000 range) and those players know how to play her well!
After many more attempts, I have come up with the placement (at the Red buff), but I do not yet have a screenshot to back it up. I will attempt to explain it as best I can, and will try to get a screenshot at some point, to update the answer.
The biggest issue I was having was that I would throw the axe, hit all 3, but not be able to pick it up to lower the cooldown, without moving. What I've found however, is if you stand behind Red, between Red and the smaller vertically aligned creep, as close to red as possible without attacking, you can throw the axe to the other side of Red (so that your cursor is the red attack sword, but it is on the opposite side of Red.)
It may take a bit of experimentation, but this seems to hit all 3, and causes a reduced cooldown. The axe has a larger radius of effect than I expected, so you do not need to place the cursor all the way on the far creep.
Hopefully this helps someone. If no one else, it helped me.
Best Answer
Since your passive refunds energy always make sure that you AA twice before using another ability. Below is the rotation I usually take.
W1 >> AA*2 >> W2 >> AA*2 >> Q1 >> AA*2 >> Q2 >> AA*2 >> E1
I always find it better to start with W because the jungle camps hit hard early and the sustain from your second combo'd with smite helps you stay alive. Do not use the second E because it does no damage and doesn't reduce the attack speed of enemies any more making the spell useless while clearing. never start with Q because your second Q is an execution spell. It will deal more damage based on how much health the enemy is missing. Using it second allows you to use the spell vamp from the second W.
I hope this helps
Jaimin
Source: I am a jungle main and have been playing since Season 3.