The most significant item you should be aware of is the ability of some boards to unlock disabled cores with the 710 or 750 southbridge chipset. Otherwise, I would select a board based on your budget and case size: 770 is atx only while the 785 can be found on both atx and micro atx boards. Some cpus won't remain stable when you unlock their disabled core, and some overclock better by leaving the disabled cores alone. You'll just have to try it.
No, the onboard graphics is part of the Northbridge, which is the 770 or 785 part of the model number. The south bridge (710, 750, etc.) is mainly for IO and other tasks. So to answer your question, no, the 770 board does not have onboard graphics at all.