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.

Related Questions

If a dental implant costs a dentist about one hundred dollars why is it that the surgical procedure is three thousand with abutment?

Labor. The cost of labor(staff )and behind the scenes costs such as overhead, lab and imbursement of the surgeon for his skills,equipment Also future followup appointment costs are generally included in that figure. Read more...
Hidden costs. are also a consideration like malpractice insurance, costs of imaging systems, and other things that go behind the scenes to make that implant possible. Read more...
Just a wild guess. Because dental school is more expensive then an average house and practice expenses (overhead) and malpractice insurance, etc. Read more...
Many costs. First,any implant from a reputable company costs well more that $100.Also,the components involved to place it are very costly (Surgical drill,multiple burs,torque driver, etc).These will need to be properly cleaned and sterilized.Other instruments needed to clean-out and prepare site.Possible grafting needed. Most importantly the knowledge and skill to properly and successfully place it.Good Luck. Read more...
Many Other Costs. The implants cost more than $100, but that's not the point. Fees encompass so many other costs and also include a profit for the office. You can use your analogy for almost everything that has a price on it, but in the end the result is the same: if you want it, you'll pay for it. The choice is yours. Read more...

Most dental implant surgery results in the patient requiring a custom made abutment at a very high cost compared to one not custom made. True, right?

Dental implant. The type of abutment depends on the type of implant, the location of the implant, and the amount of space between the implant crown and the adjacent teeth. Talk to your dentist for a more informed diagnosis. Read more...
Not necessarily. There are clear indications for a custom abutment but when those are not present a pre-fabricated abutment should be fine at a lower cost. The only way to really decide this is after an in-depth discussion with your dentist to explore costs, risks, and benefits of these options. Read more...
Use Prefabricated. A well-placed implant allows dental practitioner to use a prefabricated (not custom) abutment and reduce cost to the patient. Read more...