I think I almost always hold Back or Forward when I throw with the Hooligan… I’m pretty sure it’s Back. I know 100% I’m not in Neutral when I try to throw. To be honest, I’ve never found it a concern on the throw. If you are having trouble throwing people, you might just be too far away. Most of the time I do it, it’s never really a question. I do the Fierce Hooligan Roll, spam Fierce, and I throw them. lol! I hate saying it that way, but really, I almost never try to time it unless I’m far away. The only time you really need to time it is if you do the Strong or Jab one, and as I said earlier, I use the Fierce Hooligan Roll 98% of the time.
And yes, Cross-up Short too high is throw bait. Most of the enemies, you can land with a Cannon Spike and be safe and beat their throws. 
But the real answer is that, well, just learn the timing right and it should combo most of the itme you do it. Stand Close-Up Strong might be the easiest to Combo since it hits so fast and can be Linked into other things. And once you get good enough at it, even if you succeed only 75% of the time, it’s enough so the opponent won’t be trying to throw you every single time.
As for Cammy’s Super, I will admit that 80% of the time I use it, it’s to chip damage someone to death. Because Cammy gains meter slower than any other character in the game (absolutely ZERO free Special move whiffs to gain Meter). So by the time I get it, my opponent is usually mostly dead and chip damage is all that is left. Also, using it for going through fireballs is obvious.
It’s a real pain to Combo into the Super, as well. The best Combo is Crouch Strong, Crouch Forward, Super. But it’s so easy to fuck that Combo up becuase Crouch Forward buffers into Cannon Drill, or doesn’t come out (you super instead), etc. Practice enough and it’ll get simpler, but you won’t find tooooo many chances to use it. You can also combo Crouch Jab x 2 or 3, Stand Far Jab into Super and that Combos. It’s much easier and you can react to the opponent getting hit well, but her Stand Jab goes OVER a lot of crouching people. It’s really good for after Cross-ups, since they will be standing up when you hit them most of the time. And Crouch Jab is fast, so it’s easier to Combo after Cross-up Short than other moves.
But to be honest, in all the games I’ve played with Cammy (and that’s a whole hell of a lot), I have never landed that Combo in a match-up. EVER. No lie. Too many factors going against it:
- I never did Cross-ups.
- By the time I have enough meter to try the Combo, they die after like 3 Jabs anyway.
- Landing a Cross-Up with Cammy can be tough.
- Comboing Cross-up Short into Crouch Jab can be tricky.
- The Combo is actually kinda tough to do.
- I don’t rely on it, so I almost never try it.
So sadly, my Super use for Cammy usually falls into the punish whiffed moves, chip damage to death, or go through Fireball variety.
But the real answer is that, well, just learn the timing right and it should combo most of the itme you do it. Stand Close-Up Strong might be the easiest to Combo since it hits so fast and can be Linked into other things. And once you get good enough at it, even if you succeed only 75% of the time, it’s enough so the opponent won’t be trying to throw you every single time.