IPhone – Can I buy a used iPhone 5 from eBay and use it as an iPod Touch

This means that the phone was software unlocked (which requires a jailbreak). Because you updated the device using iTunes, you have removed the jailbreak and the phone is no longer in a state of carrier unlock. This is actually common practice and a great number of people end up getting burned (they either just don't know better or are lied to and told the phone is factory unlocked).

You can call the carrier the phone is originally locked to and have them unlocked it (they may charge a modest fee or if you luck out, do it for free as that particular model no longer being manufactured) or re-jailbreak the device and software unlock it yourself, provided you have a baseband revision of 06.15.00 or older.

Note: Apple can't help in these matters.

If you use the T-Mobile SIM it will only activate the device, it will not create a contract. If you feel safe you can get it without a SIM, head over to the closest Apple Store and ask them to activate the phones using AT&T dummy cards.

Once you activate the phones, I suggest putting the phones in airplane mode and turn on Wifi, otherwise the phone will constantly be searching for signal which will cause the battery to drain.

Good luck.