The ship flying away from Rey in her flashback of being left on Jakku looks like one of the ships in the group that takes her away when she is captured.
If so, and it is a First Order ship that dropped her off, this is an interesting Easter egg.
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We don't know. As of April 4th 2016, there has been no in-canon description of the vessel matching any known ship found elsewhere in the Star Wars universe.
The ship was called the Bestoon Legacy. It was, we learn in Rise of Skywalker, owned by a Sith loyalist named Ochi. The film's Visual Dictionary identifies it as a Subpro Corporation Model WTK-85A interstellar transport
For the record, the ship doesn't match the First Order ships you're describing.