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>How does one get into trapping small game?
Traping small game is illegal in my state (PA). However coon and possum trapping is an easy and fun way to get your feet wet here. Coons taste good and their fat and fur is also useful. Check your local laws. Research your local trapping associations. I took a course last summer. 100% free and it was definately based. Got fed for a day and went home with traps, lures and all kinds of goodies. I also watch trapping videos on YT like Dave Canterbury's channel. Literally hundreds of trapping videos.
>What if I catch a non-targeted species
Use the right presentation, bait/lure and trap. Make a catch pole for releasing animals from a trap. Plenty of videos showing this on YT. Research dogproof traps and why properly sized leg hold traps won't hurt pets.
>Dispatching?
22 ammo is availible here. I also have a shit ton since I am a shooter. I use a small Ruger Bearcat or NAA mini but sometimes an old beater single shot 22 for a little more versatility. The latter can be loaded with a 22 pellet and a brown Ramset blank. I also use a 12ga to 22 rifled chamber adapter loaded with the Ramset/Pellet. A club or knife lashed to a pole works too.
Have fun outside. Start with just a trap or two.