Couple things.
Close GP, NJP, 1(x3), 12xxdash punch 28%
Close GP, NJP, 4, 1(x4) 12xxgotcha grab 35%
Doing NJP actually does a bit less damage than just doing another 1, and I personally find NJP slightly more difficult.
Anti air 1, 1(x3), 12xxdash punch
Air-to-air punch, 1(x3), 12xxdash punch
When starting a combo on an aerial opponent, the first hit does not increase the opponent's downward acceleration, so you can do an additional 1 with little to no more difficulty than the usual jab juggles. The same goes for punishing opponents that spin in the air after a blocked special.
Punishing Mileena's EX Teleport Kick is a bit more difficult... I usually block and do d+1 > (1)x3 xx Gotcha Grab, but I just judge what I can fit in based on the height and distance from me at the time. Some moves that leave the opponent spinning in the air on block can be punished by mid Ground Pounds, such as Raiden's superman, but I haven't experimented with which other ones it works on, and it only does about 2% more damage and builds an almost negligible amount of meter.
To punish fast teleports such as Raiden's, I usually just d+1 or 1.
In the corner, I use:
f+4,1,3 > d+1 > (1)x5 > (1,2)x2 xx Gotcha Grab (omitting a 1 or 1,2 if I feel I might drop it)
b+1,2,4 > (1)x6 > (1,2)x2 xx Gotcha Grab (omitting a 1 or 12 if I feel I might drop it)
f+2,2 xx Dash Punch > d+1 xx Dash Punch > d+1 xx Gotcha Grab (Difficult, but even dropped does the same as or more damage than other options I've found and doesn't throw the opponent out of the corner... still don't use it very often)
Ground Pound > dash > 4 > (1)x5 > (1,2)x2 xx Gotcha Grab
f+4,1,3 > d+1 > (1)x2 > 1,2 xx Overhead Smash > EX Ground Pound...
b+1,2,4 > (1)x3 > 1,2 xx Overhead Smash > EX Ground Pound...
Ground Pound > dash > 4 > (1)x2 > 1,2 xx Overhead Smash > EX Ground Pound
...EX Ground Pound > (1)x3 > 1,2 xx Overhead Smash > EX Ground Pound...
...EX Ground Pound > (1)x6 > (1,2)x2 xx Gotcha Grab
As for anti-airs, in addition to what OmegaK said, don't forget about Jax's 7-frame uppercut, which is particularly useful when you want to avoid switching siddes with the opponent using anti-air grab. Also, strange thing about the anti-air grab is that Jax is completely invincible, even avoiding damage from projectiles, (I believe from startup to first active frame, which is 15 frames in) as long as the opponent is in the air, so it can be situationally useful in that regard. Also, keep jump kick xx Backbreaker in mind.
As for being safe on block, you don't really have to worry about it. All Jax's specials are safe on block, except perhaps a close-range Energy Wave(I'd have to double-check that). Most of Jax's strings are safe on block... I think f+2,2,b+1 and 1,2,3 might be unsafe. f+4,1,3 gives enough advantage on block for a d+1 in most cases. If you want to make a string safer on block and you are fast enough to recognize that the opponent is stand-blocking, you can cancel into Gotcha Grab on a standing opponent, which is neutral on block. I find it particularly helpful after a blocked f+2,2, which gives plenty of time to hit-confirm and observe the position of the opponent's block. I also use 1,2,3 xx Gotcha Grab (the grab gets blocked) or 1,2,3 xx Overhead Smash to punish blocked specials that leave the opponent standing, but its effectiveness has been significantly reduced since the last patch that removed the around +10 frames Gotcha Grab gave on block. (I still find it useful, though).
Also, while CD Jr and Tyrant are good to watch, I'd try to avoid watching some of their pre-patch videos, or at least recognize that some of Jax's options have changed significantly if doing so.
