Transmission jump ( kind of like a THUD sound???)

Sounds like the electronic detent is acting up. Take it to a reputable transmission shop.

No idea without a drive so please ask a local mechanic to have a look for you

diff noises are a harmonic whine on drive (Acceleration or drive speed )or a growl on overdrive ( slowing down or coasting)
if you have a drive shaft to the rear diff it is possible that the rear shaft has not been assembled correctly and the out of phase of the joints will give indications that you describe
Take it to a drive shaft specialist shop for a diagnosis

orbital clutches slipping , needs stripping

sounds like new transmission time. im sorry.

first thing change filter and put Lucas trans non slip in it. other than that it will need a rebuild.

throttle cable

2008 Tucson? It should still be under warranty. Does it have more than 100,000 miles on it? Are you the original owner? If not, does it have less than 60,000 miles on it?.. If so, it's still under warranty - take it in to get fixed under warranty.

at that high a mileage it could be very likely the trans is starting to fail, have the computer tested for transmission trouble code's (trans is fully electronically shifted and controlled), also it is very unlikely the alignment check is related.

its kinda hard to tell what the exact problem is with all the***'s but it sounds like you might have a loose gear in your trxmission, need a little more info tohelp you on this one but also check the transmission fluid and maybe get it flushed and replaced if you have anymore information please let me know and i will get back to you as soon as possible