Basically all are relatively easy to care for. Typically mice are a bit more jumpy in regards to being held unless you raise them as a baby and hold them a lot. There was a gerbil in my classroom as a teen (quite a few years ago) and I really don't remember much about them. Honestly as a pet, I would go with the hamster. They are relatively laid back and can be handled very easily.
As for a cage, a standard hamster cage that you can buy at a local pet store will work fine. Either the wired one or the plastic one will work, but know that the plastic one has the chance of being eaten! Both mice and hamsters will chew on the plastic and could create a sizable hole to climb out of. Some of the newer cages are a metal and plastic mix with the main area being metal wire to prevent escapes.
As for the number of them, I would suggest that since you don't seem to have a lot of time to dedicate to them you should get two so that they have social interaction. Obviously, get two females or two males so you don't have to worry about them breeding.
None that you have asked about really need out-of-cage time as long as they have a cage that is large enough and well suited for them. As I said earlier, many of the newer cages are great, have lots of running room, tunnels, etc for them to play in. Definitely purchase a wheel for them as they will spend countless hours on it. Invest in the non-metal wheel...trust me it's worth the extra cash...when you don't hear a squeeking wheel at 2 a.m. you will thank me! :)
Definitely secure the cage in a location the dogs can't get to. I breed mice for food (snake) and I have my mice in a 20 gallon aquarium and my dog loves it. She will sit for hours just watching them run around. But, sometimes she gets a little excited and will nose at the glass, so if it were a smaller tank or unsecured, she could knock it over and I would have 20+ mice running around my house! :)