How can I login iChat with my Apple ID?
If both people are using iChat, all the features will work over Jabber, but you are out of luck if both people are not using iChat.

The MobileMe/.Mac sign in, when I used it anyway, used AOL’s OSCAR protocol used for AIM and ICQ (you should still be able to sign in with an ICQ account by treating it as an AIM account) due to an old and separate deal between Apple and AOL, so MobileMe users can talk to AIM and ICQ users without a gateway and vice versa. Still, even talking to them, the feature ratio between AIM clients and iChat is not 1:1, so some things would probably still break, iChat Theater if nothing else anyway, unless the other party was also using iChat.

So for a full featured iChat experience, send free Macs to all of your friends. Instant Messaging clients have shaky interoperability at best beyond basic text messages, which is unfortunate. As an alternative, I can recommend Skype which works on multiple platforms, and I find it also has better audio quality than iChat (both are quite crashy for me with audio chats and screen sharing though). There is also FaceTime which IIRC does use your Apple ID, but will only work for people using Macs, iPhones and iPads.