What's the point of a crypto cold wallet?

Hi guys!

I have been thinking of buying a cold wallet, like any type, and after a lot of research, I still don't really understand the following things:

Where is the private key inside a wallet, why I don't see it? Why I don't have to sign transactions with it? (In every wallet type)?

What does it mean that a wallet holds the private key in offline and in online (the main difference between hot and cold wallets)? Does the hot wallet shows everyone the private key because it is online or what? Or how does it work practically? Because I have never seen the most important part of the wallet: the private key. We are just talking about the unseen, the unknown.

And how they could hack the hot wallet easier and the cold wallet is only when I sign a transaction? How do they get the private key, with an example please (Because "hack" is not exact).

And what if I sign the transaction with fingerprint or with some code? Then how they could hack my fingerprint? And how this is connected to the private key, while I'm giving my fingerprint or code? Then it makes no sense if there is a private key and if it's online or offline stored.

How they could hack my wallet it they know the seed phrase? Where do they use them? In a wallet providers recovery site? And if so, what happens if my wallet is not connected to that specific provider? And what would happen if I would give a random set of words? I would hack someone's wallet? And how could I got access? I don't understand.

Why these companies have the right to make wallets on the blockchain, and why I can't make? And then what happens if that company goes bankrupt or something like that? How could I get access again?

Is it really safe to make a wallet (like any kind), if the instructions show me the seed phrase and in a paper wallet, even the private key (that's the only time I see the private key, but then how do I spend it? With a third party again. Then why would a paper wallet be safe If I give the private key on the screen?) while connected to the internet? And does the company see my private key and seed phrase? Because then it's totally not safe.

Im completely dumb to this, because I see everywhere that the hot wallet is online (so stores the private key, which I never see, online) and a cold wallet is the opposite. Some other information, but no further explaination for practicality. And of course I don't want to buy something I don't know.

Thank you guys for the answer in advance!