Is there such a thing as a dental implant that comes premade with an adjustable abutment head?

Sort of. Some implants for very small spaces are one piece with their abutment. These abutment sections can be ground down a little if needed. Most other implants have stock abutmenhts that screw into place. These are of different sizes, angulations and shapes. There are also custom abutments that can be machined or cast if the anatomy is such that nothing off the shelf will work.
Yes. Dental implants are restored with a component part called the implant abutment. This is the piece that connects to the actual dental implant (usually via a small screw). These abutments can be custom made abutments or they can be 'stock' abutments. The biggest difference is cost and biomechanics of the restoration. I recommend custom abutments so that your tooth is made to fit your mouth best!
Many types. There are many different types of implants and prosthetic components on the market. Your surgeon and restorative dentist will use a system that they are comfortable with and that will satisfy the requirements of your reconstruction. Many times custom work is indicated due to implant angulation and existing space. Allow your dentist the latitude to make the best restoration s/he can.
Yes, but why? The best abutment is the custom abutment, where it fit the best with good emergence profile and proper adaptation for the implant crown support. It should be zirconia or nitric oxdide coating (look like gold plated) for better esthetic. Premade abutment are usually stock, generic, and ugly (in my standard).

Related Questions

Dull pain 6 days after dental implant. Jaw very sore. Have taken anti b and solpadeine? Abutment stage. Replacement for implant that failed after 3 yrs

Surgery. Removing and replacing an implant is a surgical procedure, and will take time to heal. This is especially true if surrounding tissues were infected. Call theDentist or dental Specialist who did your surgery and report your status. Reevaluation, observation, additional treatment, antibiotics, analgesics, etc. may be advisable. Read more...
Pain after implant. Call your implant surgeon for a post-op exam. It is important to follow this closely since this is a replacement for a failed implant. Read more...