Did I pick the wrong bike?
Whether the cruiser style bike is suitable for the 'mountain bike trails' you are riding on really depends on exactly what those trails are like. The cruiser has pretty big tires so it should be able to handle rough surfaces just fine. If you are riding those trails and it feels OK, then it's OK. 

Cheap bikes from big-box stores with Shimano 'Tourney' level components are not known for durable and reliable drive trains that will work every single day. If you want to try to make it work, learn how rear derailleurs are adjusted, it's not super complicated. Basically there are upper and lower limits that stop the derailleur pushing the chain off the sprockets, and an indexing adjustment that positions the chain correctly for each gear.

To address the issues you are having, have a bike shop check your derailleur cage alignment. The derailleur hanger can get bent which messes up how the derailleur feeds the chain onto the sprockets. This may be why you are can't shift into all gears and the chain is coming off.

Lastly, don't forget the shift cable and housing. If these get dirt inside them or damaged they can prevent proper shifting.

If you are going to get a different bike, invest in one from a proper bike shop with better components, or look for a better quality used bike.