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Shadowdrgn
Haha, that list doesn't look too different from mine, sans my not being overly fond of Kha'Zix, but I've never been heavy on playing assassins.
If anything, it's easier now with Jax. (uhh rambling ahead/text wall, sorry about that.)
His first clear is rough without a good leash, but he's otherwise really no different from before. Typically Jax starts counterstrike/e in jungle anyway, so you mitigate a fair amount of damage by simply not being hit, and then stunning jungle mobs. You
can level 2 gank with him, (requiring you to typically go e -> q,) but I personally find this painful with Jax, and so typically I gank at level 4 after e -> w -> w -> q.
(I would say level 2 ganks are only feasible on Jax if your lanes ALSO have some form of CC and decent burst, especially with new jungle doing a bit more damage than before, so you're possibly coming in to gank from 75% health. But cheese level 2 ganks are still very possible..)
Typically if things are going pretty slow, you can just clear the jungle twice before needing to back at all, but of course what's going on in the match effects that. I've actually been able to jungle Jax with ADC masteries and runes and no leash in a particularly horrible game where league refused to switch my masteries, though I wouldn't recommend that.
My typical path on Jax now starts me at krugs (Red side). The stun on auto from smiting them works wonderfully with Jax's attack speed passive, and lets me rush to clear red with very little damage done to me (especially with counterstrike.) Alternately, he can start at Gromp and use the poison to speed-clear jungle. The poison damage done to mobs attacking still applies even if Jax is 'dodging' attacks due to counterstrike. Starting Gromp is a bit safer (and means I will have smite back up to get health back at red brambleback) but it's better to start Krugs if I think the enemy might be wanting to steal my red. If I'm jungling against a Shaco, or someone else with insanely fast clear and steal potential, I'll actually start
at the buff. Otherwise clear time is better starting Krugs for stun or Gromp for poison.
If the game permits, I still tend to rush a trinity on Jax before getting any sort of sustain for the jungle, since he typically doesn't need a BotRK until team fights start. If your team isn't needing many ganks, doubleplusgod; Jax still has a speedy clear and the 'free sustain' built in to jungle items is fantastic now.
That said, I typically go ranger's on Jax. He really isn't a counterjungler, and is vulnerable if caught out, so poacher's doesn't work very well on him. Logically skirmisher's should be decent on him, but I've found it's one of those things that doesn't work as well in practice. Same goes for stalker's blade. If Jax can't catch someone with counterstrike and leap strike alone, he has no business chasing them.