A lot of good advice although I wouldn't recommend a 17 HMR for deer. You may want to try cocking a crossbow first before purchasing one. A buddy has a shoulder injury (got shot several times in the shoulder with a .22 while on duty and can't bow hunt anymore. Cocking a crossbow may or may not aggravate your shoulder so it would be wise to find out before you buy.
For a coyote, have you considered a 223, a 17 centerfire or even a 17 WSM (rimfire)? All of them are little to no recoil and would be good for coyotes (obviously the 17 WSM at shorter ranges). If you were to limit yourself to 50 yards on coyotes, you could use a 22 mag with good shot placement.
Your state probably requires a certain minimum caliber for deer hunting. You should ascertain your state's requirements as that may partially answer your question. If you go the AR15 route with a 300 Blackout upper, you could always use it for 5.56/223 as well. If you need to step up to 243 for deer, have you considered one of the recoil reducing systems available? They're fairly inexpensive. I bought a slip on pad for a 45-70 that had a length of pull that was too short. The pad fixed that and also reduced the recoil considerably.
Hopefully some of this is helpful. Let us know what you figure out and how well it works.