Bolt actions may be more accurate than levers on average. But for shots within 150 yds, just about all of the levers out there will be accurate enough for shooting deer. Buy the action type that you like, and make sure the rifle fits you well. Many guys buy whatever everyone else has, and start off going down the wrong road. Pick up a bunch of rifles and action types and see what feels best to you. you should be able to cycle the action comfortably without taking the gun off your shoulder. I would really consider buying a used rifle in good condition.

For a 35 cal, I would look for a used Marlin in .356 Win (Hard to find) or 35 Rem (new or used). Browning makes a lever in 358 win. Of those 3, the 358 is the winner in my book, but any of these will suit your needs. The Browning will be more expensive, especially for a new one.

The 350 mag is a great round, never sold well for some reason. If you plan to handload it could be a good choice. The 356 can be tough to find ammo for too. 35 Rem and 358 Win should be available, but of those 2 the 35 Rem will probably be easiest to find.