The proper way to address them in English would be: Dear Mr. Somesuch and Dr. SoSo or simply: Dear Sirs, This last one is what I think works best.

I would use "Dear Dr. ______ (clinic director) and Dr. _____ (program director), I don't know if there is a rule for this but I would write the person with higher authority in an organization first. It is also possible to use a title they both share such as "Dear Professors,"

I think your best option here would be to say something like "Dear Colleagues...."

In your situation, I would first address the person to whom the email is most relevant. For example, if the email is about an aspect that the younger program director oversees, then address him/her first even though the clinic director is more senior. If the subject matter is of equal relevance to both individuals, I usually address the person I work with more often first. Alternatively, it is perfectly acceptable to address the more senior person first if the subject matter is of equal relevance to both and if doing so is your preference (i.e., more compatible with your cultural sensitivities). What is less appropriate (in my view) is to address the senior person first when the email is more relevant to the younger person. This is just my own personal preference. You certainly can write "Dear all," which is what I often do… :)