Alot, if not most new handgun shooters are thoroughly discouraged by the blast and recoil from a .44 or .45. Full blown .357 loads also will get your full attention.
A .357 at handgun distances is certainly capable of dropping deer with the right loads, and several manufacturers make fine lever actions for them.
The .357 will be more inexpensive to practice with- more shooting time= better shooting. The .357 offers anything from .38 wadcutter target loads to heavy 180 grain (last I checked) full magnum loads, a very versatile gun. Many of the old Alaska trappers carried a .22 as a side arm, and used them on everything from bears and wolves to partridge for the pot. Like some one else mentioned, after the noise of the first shot, generally any critter is going to high tail it out of there. The more compact and easy to carry the arm, the more likely you are to have it and shoot. 100 rds of .22= 4.00. 100 rounds of .38/357 45.00 (last time I checked... I reload).