1. depending on your vehicle, you could put it inside or hang it on a bike rack... some even put them on little mini-trailers. Folding trikes can go in virtually every kind of car but you have to deal with the assembly and dis-assembly on both ends. Transit in Portland OR still frowns on trikes.
2. stability is a mater of experience and how you load the trike. Smaller riders probably have greater confidence. I dumped mine... and almost dumped mine and that was enough to learn what NOT to do. On a nice smooth sweeping downhill... I just let it roll with very gentle fingers on the controls.
3. I am no less comfortable on my trike over a bike. I get a lot more respect from cars on a trike, be it by a smile or a single digit wave. I have a flag, bright triangle and light on the back of the trike and it can be seen as long as you don't surprise someone. In parking lots and going past parked cars, you need to be assertive and seen. The flag will take care of most parking lot visibility (don't fly a flag too tall!) and taking the middle of your lane where cars are parked will ellinate being door'd. Again, this is much like a normal bike. Be seen, be heard.
If you got the itch, start your quest by riding everything you can on three wheels (Shoppin'!)... If you are afraid of the itch... run away... run FAR A W A Y! These things are addicting; you've been warned.