GeneM is right. You will need to show a document that will allow you to enter your destination country. If you are flying to your homecountry, then you need to show your passport as proof that you will be allowed to enter there. If it is not your homecountry, most countries still want to see your passport and will not let you in with just the Green Card, which means the airline will want to see your passport too.
As for the different names, if it is just the different spelling, it's not a problem. My name is spelled so differently in my GC and passport, that very few people recognize it as the same name at first, and it has never been a problem. However, if it is not the spelling (for example, your passport has your maiden name and the GC has your married name), you do need to show proof that there is a legal reason for that (like a marriage certificate). The airline ticket would normally have to be issued in both names, but some airlines will not pay much attention to this (check with them in advance to avoid problems when traveling).
It is also true that you don't need to fill out the I-94 when returning to the US.