If you want to do more internships, I suggest you find some that have a shorter duration and apply to another internship once you are finished with your current internship.
Some companies offer summer internships for example, which are usually 3 months long. Obviously, they happen during the summer and you might not find such short ones outside that calendar period.
If you apply and get a long internship (6-12-more time) then I would strongly advise you against doing "internship hopping". Employers dislike this because it isn't any different than "job hopping". They invest some time in the internship program, recruiting interns, training them, etc, and they want something back out of it, be it cheap labor, or supplementing personnel to do some activity sooner, or for the most part to evaluate potential employees.
If you apply for a 12 month internship and stay 3 months then you will immediately be flagged as a no-no for employee material (i.e. if you can't stay in a one year internship how will you stay in a full time employment with no determinate work period?)
So the answers to your questions are:
Will potential employers dislike this?
Is this something that is normally done?
No, because of what I mentioned above. It doesn't send the right message and doesn't create the right relationships between you and a potential full time employer.
Do internships have certain "seasons" (ie around the time diplomas are given by universities) and could they be more difficult to find after those seasons?
Yes. Internships often happen around graduation time, and for people that are at the beginning of their career. If you've already had a job for example, some companies might not consider you for the internship (you are either an employee or an intern; if you already had jobs they will consider application to an internship as somewhat of a step backwards; the progression is usually internship then job, not the other way around, unless you are changing careers or something).
In conclusion, if you want more internships, try to do more shorter internships. Otherwise, try to find a longer and good internship, where you will be doing actual work in diverse settings, instead of an internship where you are just a "cheap worker drone". And stick with it for the entire duration.