My opinions only, and many will disagree I'm sure.
First off, I am not an expert on NY guns laws and haven't been in NY in many, many years. BUT, I believe that rifles are limited to 10 round mags or capacity. Also, you may be restricted by the "assault weapons" issue, but your original list should not be a problem.
Many "experts" on self defense state that a person must practice shooting weak side with pistols so that if their strong side is injured or wounded, they can still protect themselves. Yet these same experts are by and large quick to recommend a pump shotgun for self defense. What happens if one arm/hand or the other is injured and you can't rack a new round into the chamber? Do you want to depend on the Sarah Connor (Terminator) trick? This is the reason that a few years ago that I traded in my pump shotguns for a Benneli M4 (12g) and Mossberg tactical (20g).
I have the same concern regarding lever action rifles for self-defense. John Wayne and Chuck Connors (The Rifleman TV series) made it look easy, but that's TV.
On my land in North Carolina I see an occasional bobcat, coyote, fox, etc. A .223/5.56 round should easily dispatch any of these critters, with a big coyote having a max weight of about 40 pounds. And having owned an early model Mini-14 since the late 70's (and I'm sure that Ruger has improved their accuracy), I have found it to be accurate enough for your uses out to 100 yards, maybe more. I can't remember a misfeed or an ejection problem. It's now about 40 years old, and it's not quite as good as my newer AR or bolt action .223's, but it's good enough at least IMO. And as I stated, I'm sure that the newer Mini-14's are more accurate. Shooting deer, well in NY state, it appears that you can use any centerfire rifle. So legally you're okay with a Mini-14. Federal and other ammo makers have .223 rounds that they market for medium size game. Only you know your shooting abilities to get a good chest, neck, or head shot on a deer.
So, it's hard to make one rifle work for everything. But I'm very happy with the mini-14 for the uses you've described--and a lot of people will disagree. If you get a mini-14, get only Ruger magazines. I have found that several after market mags are not worth the metal they're stamped in.