Hello

I'm just beginning to educate myself on cryptocurrencies, blockchain tech, bitcoin, etc. and have a very rudimentary understanding of everything. Mining is one concept I just can't seem to wrap my head around, no matter how many different articles I read at all different levels of explanation/understanding.
Hopefully someone can clarify and answer these questions in a way that finally makes sense to my brain

As I understand it, in extremely rudimentary terms, "mining" is the process of verifying transactions on blockchain by solving a complex mathematical puzzle (nonce value). The first "miner" to solve the puzzle (target hash) adds the transaction to the blockchain, and (once a majority of other miners verify their answer) receives a set amount of btc.
What I don't understand, is where the "answer" to said mathematical puzzle comes from, how it is found by a miner, and how other miners can verify that answer, without first knowing the answer themselves. What determines the "target hash," and what confirms an answer to be "correct," (or at or below the target hash)? Furthermore, how does answering that puzzle confirm a transaction to be legitimate?
Finally, if bitcoin mining is the process of verifying transactions, how did Satoshi Nakamoto mine the genesis block? Obviously, without yet having any created bitcoin, there would have been no transactions to verify.
I'm thinking I may just have a fundamental misunderstanding somewhere along the process, and none of this will make sense to me until I find and correct my misconception.
Can anyone break this down in another way that might finally make sense to me?
Edit: BONUS QUESTION -
I've read some bitcoin skeptics who suggest that the blockchain will become insecure once all btc has been mined, as miners will have little incentive to continue mining once they are only receiving transaction costs as a reward.
What is the counter-argument to this? Is this a scenario Satoshi Nakamoto considered?
Thanks
