No experience with the 770. I'd stick to the 700. Don't know about the parts interchangeability between the 770 and the 700, but parts, accessories, and aftermarket support are much higher for the 700, and it has a much more established reputation.
And I would stick to the .308. Either cartridge is more than adequate for deer and coyotes. For coyotes, something like Hornady's 110 gr TAP load should be devastating. For cheap (relatively) plinking, SHTF, or possibly to control damage to pelts, the 147 to 150 gr FMJ load should do the trick. Any 150 to 165 gr JSP or other hunting projectile should work very well on deer sized game, with the tougher 165 to 180 gr bullets being adequate for elk and other heavier game if the option ever becomes available to you. And there are lots of 168 to 178 gr OTM rounds available to help you get the most accuracy you can from the system on the range.
You'd really have to watch your backstop and be aware of the potential for ricochet, but for simplicity, I would probably stick to the 150 gr weight using two loads for pretty much everything--the standard GI ball load for coyotes, practice, and civil disorder/zombies/Red Dawn/ect., and a 150 gr expanding hunting load such as a Hornady Interbond or SST, Nosler Ballistic Tip, or the like for game such as deer. This would allow you to avoid having to learn multiple trajectories and zeroes for you rifle and optics.
