After reading your post I am compelled to ask if this is your first MMO game? As far as I know, all AAA MMO titles require a monthly fee to play. The first 20 levels or so of the game are in the starter area of Tortage. There are two ways to play here ..... Daytime which is like the more common MMO multiplayer scene, and Nighttime which is a totally private instance version of the area for single player play. You access the nighttime through your trainer once you get into the Tortage city. Even when playing during the daytime you can still solo the missions (if you are of the proper level to survive) but you will see all the other players running around.
After lvl 20, or you leave the Tortage island the real game starts. There is no longer any option to pick night or day ... as there is no way to just single player play by yourself. Now this does not mean you can not do the missions by yourself, but like daytime in Tortage you will be seeing the other players running around. The Dev's have stated that it is possible to go all the way to lvl 80 and not group with others (essentially solo playing), but I have to assume that there are some quests / raids that a person who chooses such a play style will never be able to experience.
After saying all that, you will still need to pay to play this game, either through a credit / bank card, or buying time cards from your local game store. I believe even when the game gets released on the console systems there will be a monthly fee to play the game. That is one of the issues they are currently in discussions about while working on how to get it on the game consoles.
So, after the free 30 days you get after buying the game, if you do not have a way to pay to play (and you have to set this up when you first make your account) your access the game will end. When you start paying for the game though, you characters will be there waiting right where you left off, so no worries about having to restart from the beggining again.
I hope this helps explain your questions for you, and I hope you get into the game. I have been in the 3 day early start program and the game is awe inspiring. I am totally loving it, and I havent gotten past lvl 17 yet.