Yes and yes, although in most parts of the world it won't be legal when done via antennas. You can buy blank SIM cards from vendors like Sysmocom, which are preprogrammed but "writable" SIM cards. The important part is knowing the private key that is used to authenticate end user devices.
Then you'll just need a decent SDR and it actually works fairly well for small test setups.
Nitpick: in most places you need a license to do this with antennas (in some places temporary licenses aren't that hard to get for temporary/noncommercial use)
Do you have an idea what would be my external IP address? On my phone connected to a mobile network, I get assigned mobile IP address which is my external IP address. It's not attached to the SIM card because it changes when I reconnected. Is it handles by the BTS software? Do I get assigned an IP address and BTS communicates on my behalf using that address which comes from the mobile network operators pool?
The LTE core assigns an IP address to your SIM card/UE. In the case of srsRAN, there's a simple CSV file mapping the IMSI to an IP address, but there are no limits on how complicated it can get.
Then you'll just need a decent SDR and it actually works fairly well for small test setups.