[G2 General] ChrisGermany2017:
Edit:

Ah that make sense. The IMEI is stored in the EFS File System. Not sure about the flash steps of a Galaxy Note 2, but that sound that you should backup a working EFS and Restore it to your working ROM.

That issue can also appear if you(r phone) did not mount EFS. The Way with that copy (Backup) and restore (apply that backup). Should work. (If your original EFS-File was not corrupted at the Time you taken that Backup).

When you wrote that everything works fine with your Stock Roms Backup you could solve that by taking another backup of that working EFS-Partition/FILE, an apply that to your phone.

Take care that the partition should be mount like an USB-Drive in your Bootloader (TWRP?), its like an usb drive, if you did not attach it during the use you cant write or read it. Not sure how that works on your phone could be just a folder or an extra mount able drive.

I am not familiar with your phone, and the Data of that drive. Should be important that you use that backup and not just set your phones IMEI manually. Just in case if nothing else works that can be an option.

Try to search about that EFS Stuff i am sure you can find someone who can describe it better then i. Can't you help efficiently without that phone, otherwise i would write you a step by step guide.

End-Edit

Then its no Hardware issue. Should be your ROM, you could try a different or to talk about it in your ROMs Community. It can be some App disrupt that too but i think thats not so likely.

On my LG G2 with lineageos 14.1, i can find my IMEI (Simcard inside) with the special dial and in: Settings > the entry "over the Phone" > Status > IMEI-Information > IMEI.

However. If your Stock backup work write down your IMEI and maybe this thread could help. However if you have no MTK-Chip (MediaTek), it would not.

Another solution, not sure if it works today because its from 2013 could be: Change imei note 2

But be sure that you read the other posts in the linked threads about that it might be illegal if you use not your or an old ones broken phone imei.

I think you should try your card another Simcard in your phone. Try to Chance the Network-Provider Carrier*** or just another ROM (that one that you used could just have a Bug in that Version.).

The Providers use the IMEI to block stolen phones... when you say it work with your Stock... oh! You don't have a Locked Simcard? Years ago some Provider sell 300 Euro Phones for 1 Euro and bind/lock them to a Simcard. To Force them to use the Phone and there contract. Not sure if that mechanic could be the source of your problem. Normally that practice is extinct. Usually it should just avoid to use a funding Phone with a cheaper providers SIM-Card.

***

Settings > Wireless and network > Tab Mobile Networks

The Phone show all Mobile Networks in your Range, you could Choose one to connect.

However without IMEI this should not help, but if your "No Service" issue is just a missing Mobil Provider or an Autoconnction to a wrong one. That could be the issue. Not sure if it will coast more (money) if you use competitive Networks to your Provider, but if that work, i am sure you can select your contract Mobil Provider that way too. Short: If you access that manual setting the phone search and try to adjust that manually, sometimes that helps.