Does holding an MEI entitle you to instruct in ASEL/ASES as well?
No. See FAR 61.195 (b):

Aircraft Ratings. A flight instructor may not conduct flight training in any aircraft for which the flight instructor does not hold:
(1) A pilot certificate and flight instructor certificate with the applicable category and class rating...

Without an Airplane - Single Engine rating on the flight instructor certificate, an instructor may not conduct flight training in a single-engine airplane.
As an aside, a CFI/I/MEI needs only a third class medical. See FAR 61.23(a)(3)(iv):

(3) [A person] must hold at least a third-class medical certificate— ...
(iv) When exercising the privileges of a flight instructor certificate and acting as the pilot in command;
(v) When exercising the privileges of a flight certificate and serving as a required pilot flight crewmember

In addition, note that these requirements are for an instructor also acting as a PIC or required pilot. When the student is the only required crewmember for a given flight, the instructor is not required to hold a medical certificate at all.