Can a student pilot start flight training before they get their student pilot certificate and medical?
Yes. 

14 CFR 61.87(n)


  (n) Limitations on student pilots operating an aircraft in solo flight. A student pilot may not operate an aircraft in solo flight unless that student pilot has received: 
  
  
    (1) An endorsement from an authorized instructor on his or her student pilot certificate for the specific make and model aircraft to be flown; and 
    
    (2) An endorsement in the student's logbook for the specific make and model aircraft to be flown by an authorized instructor, who gave the training within the 90 days preceding the date of the flight. 
  


14 CFR 61.23(a)(3)


  (a) Operations requiring a medical certificate. Except as provided in paragraphs (b) and (c) of this section, a person— ...  
  
  
    (3) Must hold at least a third-class medical certificate— 
    
    
      (i) When exercising the privileges of a private pilot certificate; 
      
      (ii) When exercising the privileges of a recreational pilot certificate; 
      
      (iii) When exercising the privileges of a student pilot certificate;
    
  


The student pilot only needs a student pilot certificate and medical when operating in solo flight.  

With that said, if there is any doubt that a medical certificate won't be granted it is a good idea to get that out of the way before spending money on flight training you will never be able to use.