I have two bathrooms with towel rings in that location (and toilets to the left) and both have outlets in similar locations. I took a folded hand towel (to simulate looping over the ring) and hung it so the towel hangs just above (maybe .5-1") the countertop. I really like having the hand towel there. It means that no one is crossing the room with dripping hands or potentially dropping a towel in the toilet (if I hung it over the toilet instead). I don't think there's a right or wrong answer, it just depends on what looks and functions well for you.

Paper Towels: On a vertical stand next to my sink. I like having them handy. It is an inexpensive one from Target. In the past, I had them hanging under the upper cabinets. After we remodeled, I needed a new solution. I didn't think the countertop one would be stable, but it is and I really like it. I enjoying being able to grab the whole roll and take it to another room if I need to. Wet dish towels: I have a Rev-a-shelf hanging rod thing on the inside of one of my cabinet doors under the sink. I only use it occasionally though. Usually I have a dishtowel folded in half laying/hanging right on the counter where I stand at the sink. I use it for drying my hands, fruit, dishes, etc. throughout the day. It dries quickly. I have a dish drainer on one side of my double sink, so anything I hand wash usually just dries in there. If I've entertained and have extra hand washing with more dishes on a towel on the counter, then those go on the racks under the sink until they dry. Sometimes if I have one that is really wet and yucky, I just throw it on the tile floor in my laundry room (closet). It is the only tile I have in the whole house (I like to take advantage of it.) :) Cutting boards: In a vertically divided cabinet in one of my lowers. Plastic bags: I donate them to a food bank, so I throw them in the trunk of my car until I see my friend who takes them for me. :)

For thirty years, when I had one too many wet dishtowels, I threw one over the top of the door to the hall, a veneered hollow core oak door that stands open most of the time. If anyone reading this is also a practitioner of this trick, be ready to see the veneer de-laminate right at the top middle of the door on both sides, unless you urethane the heck out of it. But it's handy!

If I were to hang my dish towel (even a clean dry one) on a lower cabinet, dishwasher, hook, etc, my dog would definitely play with it and possibly eat it. (She has opened a lower cabinet door to eat clean dish towels before.) But I personally have thought about putting a hook or towel ring or bar (like you might use in the bathroom for a hand towel) on the wall by my sink or on an upper cabinet side next to the sink. Or possibly a towel bar mounted under the cabinet close to the backsplash. I would want the towel not to be too obtrusive while hanging, but accessible. I haven't done this (since my kitchen is not remodeled), but it's an idea I'm hoping to bring to my (eventual) kitchen remodel.