My first question is what happens when i flip hindu? Will i get a caste? Will my heir? If not how can i get anyone in my dynasty to end up with a caste?
There should be an (expensive) intrigue menu decision to "become kshatriya".
According to the wiki: "Spouse and children also become Kshatriya (if unlanded and in character's court)".
Second question is if this is a good time to flip, or it might be better to abuse the muslim invasion a bit more. Main issue is decadent family draining my piety but also a few large kingdoms being shia.
If you flip and create the empire of India, you will get increased levies and taxes from vassals that are currently in bengal & deccan. But you might lose de jure drift progress on other kingdoms.
However, you will over time lose the effect of Jizya. (You'll lose it immediately on your own demesne. Afterwards, as count+ vassals convert to Hinduism, they will also lose Jizya on their own demesne.)
Third question, does it make sense as hindu to focus on full conversion or will i have to accept having buddist and jain vassals?
It takes longer to convert provinces to a religion in the same religious group. (Eg: It will be slower to convert a Buddhist county to Hinduism than to convert a Shia county to Hinduism.)
For vassals, it depends how much you want to micro-manage. You could in theory force conversion all the way down to baron level, but that sounds like a really boring way to spend 6 hours. You're probably big enough that you can ignore the "heathen" opinion malus from the vast majority of your vassals. For those that you can't ignore, I suggest you find non-religious reasons to revoke their titles - religious revocation gives a negative opinion malus to everyone of that religion in your realm, and your realm is massive, so religious revocation is really an all-or-nothing strategy.
Fourth question, how the heel do you kill the tiger with your own hands?
If you have your capital in India, the normal grand hunt event chain is replaced by a tiger hunt. You don't always find the tiger, and you don't always face it yourself.
Not related but kill the caliph/pope with elephants?
Imprison the Caliph/Pope and execute them. Maybe back up your save beforehand, because I'm not certain that the execution method is forced to be elephants.
Last question, do surrendering to a mongol invasion destroy my empire title? or would i be a vassal emperor (mongol invasion hasnt happened and i'm getting worried given my expansion north)?
If you surrender immediately, you will lose whichever kingdom they target. If you let the invasion run to 100%, you might lose some other counties in addition (depending on what they siege).
You are big enough to defeat them, but there might be logistical problems getting your armies into position if you're the first person they attack. In this case, it might make sense to: surrender immediately, stabilise your realm, declare holy war on them (repeatedly). Their event troops won't be replenished for several decades after they initially appear, so you can definitely wear them down by attrition.
Also, remember that the Mongols are very likely to win the war in China. If you get some indication that they are about to appear, you should start spending all your Grace asap.