Ok, im maining jax since before the halloween patch (last year).And i wanted to point out some things.
1. "AppleJax doesn't need to build magic resist because of his ultimate. ****in' smart." it's not true, it would work out only if the enemy team had 1 or none AP user, or if they had only non-spammers, which is pretty rare these days.
2."If you choose to focus AppleJax first in team fights, you're just giving him 'Counter Strike', if you target him last, you wont have a team left to help you." ... If you focus jax, you will totally shut his dmg output (with disables etc.) and get a free kill. As jax you generally want to sit back and when someone initiates, you need to kill squishies before they kill your tank. If you pick a jax without your team having someone capable of eating all of the dmg you're boned.
3. Items, i think that ph. dancer is not viable with your other items. Yes, crits are awsome and all that stuff but you have only about 250 dmg (extremely low for AD), and again, any AS except rageblade and ult is overkill because you usually dont have time to throw out attacks. Jax is best played as a skirmisher, thus jumping in hitting once or twice and running away (you get focused -> you die).
Jax is one of the best 1v1 champs, and at lvl 6 you kill everyone ( sometimes 1v2), but gunblade is one of the best items on jax at the moment (even though they nerfed it). As you said, jax doesnt have any regen at all thus delaying it too much is not smart (i would suggest trying to take it 1st or 2nd and you'll see it does wonders). Take the slow as a bonus*.
4. Summoner spells - at least make make Flash optional (because it's the best out there, I take it almost on every champ i play and it saves me at least 5 times a match)
Well, that should be all from me now (a wall of text) but I'm giving you -1 for now. Ofc I'm going to check it out when you update. See ya next time *Kickass out*.