Yes, they would be considered Messianic Jews.....
Following the teachings of the first church in Jerusalem...
They were Jews I think called Followers of the *Way*
Yes, they would be considered Messianic Jews.....
No, they would not. If that is their motivation, they have been profoundly poorly taught.
Paul's letter to the Galatian church would be helpful to review.
Look: I, Paul, say to you that if you accept circumcision, Christ will be of no advantage to you.
3 I testify again to every man who accepts circumcision that he is obligated to keep the whole law.
4 You are severed from Christ, you who would be justified by the law; you have fallen away from grace.