Can you have an AME also sign off on your FAA BasicMed paperwork?
From the FAA's FAQ:


  Q35: Can my AME provide the medical examinations under BasicMed?   
  
  A:
  All AMEs are state-licensed physicians and may elect to conduct
  physical examinations required under BasicMed. But the FAA does not
  require AMEs to provide medical examinations under BasicMed. It is up
  to the individual AMEs to decide whether to offer BasicMed medical
  examinations based on medical practice considerations


The key words in there are "may elect to". There have been several discussions about this on AOPA's forums (members only) and people's experience so far is that most AMEs won't do BasicMed. The only specific reason I've heard is liability: one explanation I read is that when acting as an AME, the physician is representing the federal government and is therefore more or less immune from liability. But if he does a BasicMed exam he's acting as a regular state physician, and his liability insurance may or may not allow it. So you just have to ask your AME if he'll do BasicMed or not.

As for whether BasicMed affects a third-class medical or vice versa, this is from the same FAQ:


  Q2: How does BasicMed affect FAA medical certificates? Does FAA still
  offer the third class medical?  
  
  A: BasicMed does not affect medical
  certificates at all and did not do away with third-class medical
  certification. In fact, third-class medical certification is still
  required for GA aircraft and flights that do not meet the criteria for
  BasicMed. Nothing about the FAA’s medical certificate program has
  changed with BasicMed, and you can still apply for a first, second, or
  third class medical the way you always have. BasicMed is merely an
  additional qualification you can use to fly, in lieu of holding a
  medical certificate.


And this:


  Q6: Can I exercise BasicMed and hold a medical certificate at the same
  time?  
  
  A: Yes. If you are operating under BasicMed, then you must comply with
  the BasicMed operating limitations (e.g. flying only within the U.S.
  and at or less than 250 knots). When operating under BasicMed, you are
  not exercising the privileges of your medical certificate.


Simply put, there is no link or dependency at all between an FAA medical and BasicMed. Your AME might do one physical exam and sign off both at the same time for the sake of efficiency, but that doesn't change anything about the renewal dates.