It's possible and it can be done in a number of ways:
You can file the papers yourself, or use an online service like http://incorporate.com/ or http://legalzoom.com. Their services start at around $500, I believe. Or you can hire an incorporation attorney to help you, but that will be significantly more expensive.
Either way you'll need a registered agent for legal purposes, in the state you form your LLC in. Some online incorporation services provide that for a year too.
You can incorporate in almost any state, but most people choose Delaware, Wyoming or Nevada as they are more business friendly (low taxes, less paperwork).
If you (or your company) needs a bank account in the US, you'll have to fly someone in the states to open it in person. Or if an US bank has a local branch in your area, they can probably help with that.
Yes, LLC's may have owners that are foreign entities. However, Simply filing the Articles of Organization with the Secretary of State and paying the filing fee is not enough to protect you. To maintain LLC liability protection, you should become educated with the various legal requirements and compliance issues.
Entities (LLC included) are registered at the state-level, not the federal-level. So it all depends on the state.
In my state, Ohio, I don't believe an LLC may be owned by a non-US person/entity.