Does holding a medical certificate mean you automatically have a student pilot certificate?
This answer reflects rulemaking effective 1 April 2016
Regarding the question in the title, no, holding a medical does not mean you have a student pilot certificate.  New regulations going into effect 1 April 2016 separate the student pilot certificate from the combination medical+student pilot certificate.  Students will need to apply for a student pilot certificate with their CFI or with the FSDO (or their part 141 flight school) through IACRA.  The new student pilot certificate is plastic just like a normal pilot certificate and do not expire.
This part is true with or without the rulemaking changes
Regarding the question in the body of your post, you are already a certificated pilot with a balloon rating.  You are not a student pilot, you are a private pilot.  The regulations you are subject to are to add an airplane category rating are in 14 CFR 61.63.

§61.63   Additional aircraft ratings (other than for ratings at the airline transport pilot certification level).
(a) General. For an additional aircraft rating on a pilot certificate, other than for an airline transport pilot certificate, a person must meet the requirements of this section appropriate to the additional aircraft rating sought.
(b) Additional aircraft category rating. A person who applies to add a category rating to a pilot certificate:
(1) Must complete the training and have the applicable aeronautical experience.
(2) Must have a logbook or training record endorsement from an authorized instructor attesting that the person was found competent in the appropriate aeronautical knowledge areas and proficient in the appropriate areas of operation.
(3) Must pass the practical test.
(4) Need not take an additional knowledge test, provided the applicant holds an airplane, rotorcraft, powered-lift, weight-shift-control aircraft, powered parachute, or airship rating at that pilot certificate level.

As a balloon pilot seeking the airplane category rating with the single-engine land class you

must meet the aeronautical experience requirements in 61.109(a)
must be endorsed and found competent in aeronautical knowledge of 61.105(b), 61.107(b)(1)
must complete a private pilot airplane written exam

and then you will be able to take the practical examination and have airplane single-engine land added to your existing private pilot certificate.