The first two sentences are talking about the start of a relationship. More common would be to say "start dating."
The last sentence is talking about a relationship that already exists. It would be more common to say "is in a relationship" or "has a girlfriend" or "has a boyfriend" or "is married," etcetera.
There is a small error in the 2nd sentence. It needs an "a": "get into *a* relationship."
None of your examples would be commonly used. It's the "get into" and "have" parts that are unusual.
It would be more common to say:
"She wants to start dating" or "She started dating."
If the relationship already exists, then it would be common to say:
"She is in a relationship."