Do small helicopters like the R44 require an FAA type rating?
SFAR 73 is Robinson R-22/R-44 Special Training and Experience Requirements and lists the specific training requirements for that model. Here's one relevant piece:


  (1) No person may act as pilot in command of a Robinson model R-22
  unless that person:
  
  (i) Has had at least 200 flight hours in helicopters, at least 50
  flight hours of which were in the Robinson R-22; or
  
  (ii) Has had at least 10 hours dual instruction in the Robinson R-22
  and has received an endorsement from a certified flight instructor
  authorized under paragraph (b)(5) of this section that the individual
  has been given the training required by this paragraph and is
  proficient to act as pilot in command of an R-22. Beginning 12
  calendar months after the date of the endorsement, the individual may
  not act as pilot in command unless the individual has completed a
  flight review in an R-22 within the preceding 12 calendar months and
  obtained an endorsement for that flight review. The dual instruction
  must include at least the following abnormal and emergency procedures
  flight training: [...]


SFAR 108 has similar requirements for operating the Mitsubishi MU-2B.

Finally, note that the type certificate is for the aircraft, and has nothing to do with whether an individual pilot is qualified to fly it or not.