Yes, you can ride trikes on all of the trails here. I would not, however, try to cross Roosevelt bridge with one because it is so narrow.
I don’t know of any place that rents trikes in the local area. Best place to go for trying out lots of trikes is Mt. Airy Bicycles in MD (about 45 minutes from Arlington on a Saturday) and/or Bikes at Vienna along the W&OD. I found the former to be far more knowledgeable and useful but it’s best to try out both.
Trikes are easy to ride in low traffic areas or along busy roads with good sight lines. I would not ride one in downtown DC but am fine around Arlington/Alexandria. It helps if you are used to riding around cars otherwise you may wish to stick to the trails till you get used to the lower profile.
The other thing to know about trikes is that it is hard to get hurt if you tip over given that you are only six inches or so from the ground but they will tip over if you take a sharp turn and do not lean to the side to counteract the wheels trying to come off the ground. If you are moderately flexible the leaning is easy but you need to make it a reflex. I rode my trike for the first time in two months due to the nice weather and forgot to lean when I tried to did a u-turn. Opps, when right over. Neither me nor the bike were hurt at all; just my pride. I’m sure it looked rather funny to the bystanders. Just wanted to mention that since you said you have balance issues.
Trikes are slower than regular bikes, esp up hills. They also work the muscles differently. I bought my trike from Mt. Airy last fall but am having knee problems when using it. Still need to figure out what the issue is. Other than that it is super comfortable and provides a much different riding experience than the upright.
Lastly, please note that trikes are generally more expensive to buy and equip than uprights. But they hold their value well.