You have to file Joint. She can apply for a ITIN, which is a number for people without SSN. All she needs is a birth certificate and 1 ID. This can be from her country but it cannot be expired.
Hardship Letters are written when you are going through financial problems.You write this letter to the broker, bank,or financial institution asking for loan.That means this letter is very official...
An invitation to the USA will have absolutely no affect on an immigration application.Whoever you want to come to the USA will either have to get accepted on their own merit (education visa, business...
There are specific details you must say on your letter, after which you must notarize the letter. You can find a sample letter on this website: http://www.invitationletter.net/