What Ace and LaOtto are saying I also agree with.
Well, except that Speer does make a specific 22 caliber (.224) to be loaded for any cartridge (well, maybe not the hornet) that is specifically made for deer. Yeah, I know, goes against some folks conventional thought, and you will get folk who shoot 'em with the .222's & .223's (my uncle just turned 80 and hung up his hunting....he got most with a .222, head shot's all of 'em). The Speer data...somewhere (I know it's either on the web, or in the bullet's as I have a box of 'em) tell you to stick to head or neck shots with these bullets.
http://www.speer-bullets.com/default.asp?s1=3&s2=7&s3=12But still, a .243 with a 95grain or better will do the job. And the .243 with a 70 or 80 grain ballistic tip will take out a coyote and devastate a groundhog.
I don't have the 7-08, but from the Hodgdon web site, it would appear the smallest bullet of 100 grain, if the gun can shoot 'em accurately, would take care of any coyote. And there is no doubt about it's ability on deer. (With a proper bullet)
Dave