Welcome to the forum. Initial response, based entirely on the title you chose for your question, without reading anything else: Hand-genital contact carries no significant risk for any STD, and I cannot imagine how blowing air on the penis (cold or hot!) could be a risk.
Now I have read the question itself. STDs also are not transmitted by body rubbing without penile-vaginal, -anal, or -oral penetration. And no STDs cause diarrhea as the main symptom. However, you clearly had sufficiently close contact to catch a garden-variety gastrointestinal virus. The most common such infectious agent is norovirus, which can be transmitted by very superficial contact. If you caught anything at all during the massage and related contact, that's what you have -- norovirus or some other minor viral infection. It should clear up on its own within a few days. If your symptoms persist, see a doctor or clinic. But don't worry about STDs; there is no risk.
In other words, the direct answers to your questions are 1) No, you do not need to be concerned about STDs; and 2) STDs are not transmitted by any of the kinds of contact you have listed. No testing is necessarh.
FYI, here is another thread that explains the principles of STD/HIV transmission in more detail, which should reassure you:
http://www.medhelp.org/postss/show/1119533
Regards-- HHH, MD