If finding a block is the result of mining, which is the process of transaction verification, so what will happen when all the bitcoins are mined out? The reward for mining will be gone. So who is going to verify all the transactions? will the bitcoin die when its all mined up?

Even if that takes many many years, will the halvings and difficulty skyrocketing effect the stability of bitcoin? because it will eventually just become completely unworth it to mine, which would put the bitcoin at huge risk for the 51% takeover if I am right?