Newby question regarding paying with Bitcoin.

I have to make a payment in bitcoin. And I never really dabbled in crypto.

I was wondering what a wallet actually is. And more importantly what is relevant about a wallet to me. I have a bunch of questions. I installed Exodus on my phone and I instantly had a wallet. But what actually is a wallet?

1. Is a newly created wallet with no transactions actually on the blockchain already?
2. If 1. is no; is a wallet nothing more then a public/private key combo where the public key has received (and maybe later sent) crypto?
3. Is the (public) key randomly generated?
4. If 3. is yes; isn't there a chance programs like Exodus randomly generate the same wallet as someone else already has/will have in the future (or generated at the exact same time)?
5. Are the 12 random words that seeded the private key enough to keep access to a wallet? Or do I need to also back-up the public key?
6. If I don't like exodus and want to use another program (or a hardware wallet). Can I just enter the 12 random words and/or public key to "install" that wallet in the program/device?