Can I fly in Europe using an FAA Statement of Demonstrated Ability (SODA)?
If that is not possible, then can I take an FAA medical class 1 with a
  SODA, and then move back to Europe and fly in European airspace and
  fly internationally?


Potentially, the FAA vision requirements, require you to have 20/20 in both eyes individually (corrected or uncorrected). You can find a summary of the regulations here to pass a standard medical. However the FAA does have a special issuance for people with monocular vision which you can find more info on here


  An applicant will be considered monocular when there is only one eye
  or when the best corrected distant visual acuity in the poorer eye is
  no better than 20/200. An individual with one eye, or effective visual
  acuity equivalent to monocular, may be considered for medical
  certification, any class, through the special issuance section of part
  67 (14 CFR 67.401)


This would make you eligible for a class 1 medical which would allow you to train here in the US as well as get a full FAA commercial/ATP ticket which is valid in most of the world. Bottom line is that you can become an airline pilot (at least under FAA rules) and fly international, moving back to your home country may be an issue if they don't recognize an FAA license or the airline has its own medical regulations.