Can one become a CFI with a sport license?
Yes.  

FAR 61.403.


  To be eligible for a flight instructor certificate with a sport pilot rating you must:
  
  
    (a) Be at least 18 years old.
    
    (b) Be able to read, speak, write, and understand English. If you cannot read, speak, write, and understand English because of medical reasons, the FAA may place limits on your certificate as are necessary for the safe operation of light-sport aircraft.
    
    (c) Hold at least a sport pilot certificate with category and class ratings or privileges, as applicable, that are appropriate to the flight instructor privileges sought.
  




A CFI never needs a medical certificate.  It is not required for the operation of that certificate.  

A medical certificate may be required if the CFI has additional roles.  For example, the CFI also is acting as the legal pilot in command.  A sport pilot acting as the legal pilot in command does not need a medical.  



FAR 61.23(b)


  (b) Operations not requiring a medical certificate. A person is not required to hold a medical certificate—
  
  
    (1) When exercising the privileges of a student pilot certificate while seeking—
    
    
      (i) A sport pilot certificate with glider or balloon privileges; or
      
      (ii) A pilot certificate with a glider category rating or balloon class rating;
    
    
    (2) When exercising the privileges of a sport pilot certificate with privileges in a glider or balloon;
    
    (3) When exercising the privileges of a pilot certificate with a glider category rating or balloon class rating in a glider or a balloon, as appropriate;
    
    (4) When exercising the privileges of a flight instructor certificate with—
    
    
      (i) A sport pilot rating in a glider or balloon; or
      
      (ii) A glider category rating; 
    
    
    (5) When exercising the privileges of a flight instructor certificate if the person is not acting as  pilot in command or serving as a required pilot flight crewmember; 
    
    (6) When exercising the privileges of a ground instructor certificate; 
  




To answer your questions.  A sport pilot or CFI will never need a medical because the operation does not require a medical but a valid driver's license.  

A glider pilot or CFI never needs a medical or drivers license, if I am reading the regulation correctly.  



A sport CFI does not require a CPL as does a regular CFI. The CPL requirement is found under part 61 subpart H


  Subpart H—Flight Instructors Other than Flight Instructors With a Sport Pilot Rating
  
  
    §61.183   Eligibility requirements.
    To be eligible for a flight instructor certificate or rating a person must:
    
    
      (c) Hold either a commercial pilot certificate or airline transport pilot certificate with:
    
  


But subpart H specifies Other than Flight Instructors With a Sport Pilot Rating. The rules for sport CFI are found in subpart K where no such requirement is found. (see 61.403 above)



It does not relieve the pilot or CFI from adhering to FAR §61.53 Prohibition on operations during medical deficiency though.