General
Can you play EverQuest offline?
No, you can't play EverQuest offline, the game servers are located mainly in the United States and you must be connected to them to play. When the servers are down for updates or maintenance you can't play the game.
How possible is it to play solo?
You can usually gain experience to level up your character (in regular levels and alternate advancement) without too much trouble (gets more difficult the higher level you are) but many of the quests are not possible to complete on your own. It can also be extremely repetitive to play alone as there isn't much to do except some basic tasks and killing the same creatures over and over (you tend to be more restricted where you can hunt efficiently and safely solo than in a group).
How often are the servers down?
Not a great deal (perhaps once a month) but are usually down for between 4 and 10 hours when they are (around 8 on average). This matters most if your not in America as it may be during your prime playing time. On a 'patch day' they will usually come down somewhere between 10am - 12am in the UK (GMT) and 4am - 6am in the US.
How stable are the servers?
In my opinion they are very stable but it should be kept in mind that EverQuest is a mature mmo. Rollbacks are extremely rare and (connection) lag is kept to a minimum (where it does occur it tends to be in very high end instanced zones with scripted events and many monsters).
Do players not located in America have seperate accounts?
No, all players have the same type of account and can play on any server.
Does EverQuest have International (non-American) servers?
One server is designated European.