1 - Do places like FAcebook check for duplicate IP addresses?
Facebook monitors the IP address that you log in from to check for suspicious activity on your account. So if you suddenly log in from an IP that's located hundreds of thousands of miles away from where you usually log in, it will flag that as possibly suspicious.
2 - Can anyone tell if an IP address is from a public place, like a library?
If you are connecting to a site that logs such things, they could check for that if they wanted to. IP addresses are handed out by the ISP and they are traceable.
3 - Is your IP address an attribute of your modem, or of your computer? (No smart phone in my world, so not an issue.)
There are 2 types of IP addresses. Public and private. Your public IP address is the IP address of your modem, it is assigned by your ISP. The private IP address is for only inside your house and is assigned by your router. This is done for 2 reaons. 1-- To preserve IP addresses, the IPv4 addresses that most of the internet uses are limited to approximately 4 billion, so having a single IP address for your home, and then giving private IPs to the devices inside your home means that you're only using 1 IP. 2-- As a basic layer of security. Your home computer isn't really directly accessible to the internet via the public IP.
4 - If you unplug your modem do you get a new IP address when you plug it back in?
This depends. Every time you restart your modem, it goes to the ISP and asks for an IP address. Usually your IP address will be assigned for a certain amount of time, so every time you restart within that window you will get the same IP. The only way to guarantee a change would be to keep your modem unplugged for a long period of time (like 7 days or so)
5 - Any other thoughts that might be helpful to an ignoramus like myself?
You don't need to really worry about this too much, but feel free to ask more questions if you have them.
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