IOS – No Simcard installed – iPhone 5

This is, unfortunately, expected behavior. Whenever you activate an iPhone (which you do as part of an OS install) it talks to Apple's servers and does various things like exchange crypto certs for APNS, etc. It also checks the IMEI of the phone to see what carrier Apple sold it to, and validates the SIM is valid for that carrier (if you buy an unlocked iPhone in a country where you can buy either locked or unlocked phones the vendor will actually copy down the IMEI number of a normal locked phone, and then enter in their system where it gets pushed to Apple's servers as an unlocked phone, there is no physical or configuration difference on the phones).

The easiest way to get it activated is (as you surmised) to use an AT&T SIM. You mentioned you have an iPhone 4, you can activate it with that SIM. There are plenty of companies that sell microSIM to SIM adapters, but if you want to do it right now you can just place the micro SIM directly in the iPhones 3GS's SIM try with tape. I would not recommend doing this on a device you intend to keep the SIM in for a an extended period because the SIM is likely to come loose (and jam the SIM tray), but if you are just inserting it, activating it, and removing it you should be fine so long as you are pretty careful with how you apply the tape and not shaking the phone too much.

The IMEI is kind of like a MAC address for the cellular network only. It is nearly impossible to change it, you can only spoof it on a software level. So the answer is, No, your IMEI will never change unless you really really want it to. Removing the SIM card shouldn't change the IMEI either...