Can you smart folks explain some things about IP addresses for me?

[LAS14]

A while ago we were at our vacation house and Facebook or Outlook or something like that sent me a message that my account had been logged onto from an unknown (not sure that was the word) location, but that it was "the same IP address" as you are using now... or words to that effect.

Well, I thought, how do they know my IP address? That got me to thinking about more questions, especially because, now and then I'll assume extra identities at a site like Facebook, and I know they don't approve.

1 - Do places like FAcebook check for duplicate IP addresses?

2 - Can anyone tell if an IP address is from a public place, like a library?

3 - Is your IP address an attribute of your modem, or of your computer? (No smart phone in my world, so not an issue.)

4 - If you unplug your modem do you get a new IP address when you plug it back in?

5 - Any other thoughts that might be helpful to an ignoramus like myself?

tia
las