Guild Wars cannot run offline at all. PvE is still group based, with 4, 6, or 8 people in a group, depending on the zone you are going into. You don't necessarily have to play with other people, though, as the game offers to let you bring AI-controlled henchmen of various jobs and builds instead if you want. Either way, when you go outside of town, it's just your group and all the bad guys.
When you buy a GW game, you are buying access to the PvE continent and the classes that section of GW provides. The PvP continent is included in all games, so you can PvP with/against anyone. In the game, if you want to make you strong, you should have more and more gold. You can farm by yourself, and you can also go to buy from www.ugamegold.com. This is a easy way to get it.
But in order for two people to meet up in PvE, they must both have one version of the game that they share; otherwise, they will never be on the same continent. So, for example, two people, one with the original GW (now called Prophesies) and one with Factions. If the person with Factions buys Prophesies, then they can take their Factions characters to the Prophesies continent and team up with the first person (even if their characters are using classes that don't exist in Prophesies). But the Person with just Prophesies still can't go to the Factions continent.