"I am lost" - lost is a "stative verb"; the word describes a state or condition, like "I am sick". (be + adjective = stative verb)
I lost my wallet - past simple verb
"Homestay" is actually an adjective e.g. homestay family, however, because so many students live with homestay families, the phrase is shortened and used as a noun - "I am going to my homestay". So, both your sentences are really incorrect and someone ( even an English speaker) who doesn't know about spending time in other countries and stayng with a family, would not understand your sentence.
It would be better to say that you are going to America to live with a homestay family.
No.3 Both sentences mean the same thing.
Hope this helps.
Student or Learner