How desperate is too desperate when it comes to contacting supervisor?
Keeping in touch with a former supervisor is hardly a negative, particularly since you are collaborating with him on a paper. 

At a rate of an email every couple of months, this frequency of your communications doesn't make you appear desperate or pushy. Whether these qualities come across in what you write to your former supervisor, or how you express yourself, is impossible to determine. 

Maintaining good links with your former supervisors and collaborators is excellent practice, so long as you remain pleasant, polite, and on topic. 

Deciding what the best thing to do regarding internships and applications is always tricky. Gaining the advice of your past supervisors is a good idea, and most supervisors would see it as part of their ongoing pastoral care to provide their opinion on your career progress. Don't, however, expect anyone to make a decision for you.