1. Yes, guilds are global, so it doesn't matter what world you are on. You can be a guild member, represent your guild, and participate in guild chat even if you are on another world. You just will not be able to play with your team mates while you're on that other world, and you will not benefit from any of the upgrades/boosts that they have purchased on the original world.
2. This is a "yes, but..." one. Yes, you will still earn influence for your guild while you are representing them. However, influence is counter per world, so you will be earning influence for them on that other world and not on the world where the rest of your guild is.
3. No. If you want to play together with your guild mates, you will have to switch worlds (which, at this time, is limited to once per 24 hours). At some stage in the future, ArenaNet intend to add a "guesting" feature that should make this a lot easier, but right now, you'll have to do an actual world transfer to play together.
The answer to #3 is not correct. Instances are independent of worlds, and you can enter dungeons with people in any world.
In the future, Anet has stated that they plan to release a feature called "guesting" that allows you to hop to a server that a party member is on and play with them there (in the same zone). They have said that you will keep the WvW bonuses from your original world; they have not said anything about guild upgrades, but I would assume that the case is the same in that you would only have the upgrades from your guild on your home world.