It is very possible to play PS solo, most of the combat/training is in fact done solo. Of course some creatures are very hard to take on on your own but you can definitely fight on your own as well.
PS is a game that requires some getting used to though and certain things can be very frustrating to figure out on your own, like questing .. returning from the Death Realm .. crafting .. etc. For those things it's usually easier to get some help from an experienced player. This is mainly help in the form of information though.
And of course it's always more fun to be part of a group or a guild and have friends to play with but you shouldn't be concerned about needing a party to get somewhere, you can do fine on your own.
As for your second question .. if you log out of the game you will automatically be removed from your group. I'm sure most players understand the real life issues that can appear so I doubt anyone has any trouble with it.
Am I right if I guess that you played Guild Wars or something similar before and this is making you worry? .. if it is, don't worry. PS is nothing like that.
