Do some dental implants come with an abutment already attached to the implant forming one piece that is surgically implanted at a first session?

One-Piece Implants: A one piece implant incorporates the abutment as an integral part of the implant. One-piece implants have a number of additional considerations as compared to two-piece options. Ask your dentist for more information.

Yes: Strauman was one of the first to do this as far back as the 90's. There are a few more out there now. I have been doing this for 30 years and prefer the 2 piece configuration. However, many of our restorations are screwed right into the implant , thereby not needing an abutment. Abutments are most often used in multiple tooth or full arch restorations to align or correct angulations.

One piece system: One piece systems have very limited and specific indications, so in answer to your question , yes

Yes: Yes, it is called 1 stage implant surgery... And after few month of healing, the dentist will unscew the abutment and make the impression for the crown