Yeah, food allergies/intolerances are unpredictable! They can crop up at ANY time. It does sound like an intolerance of some kind, though I wouldn't necessarily rule out allergy either. When I was trying to determine my baby's intolerances I pulled each food for about 3 weeks and then tried reintroducing to see if it was definitely that food. You might try keeping milk out a bit longer and then re-trial it.
If it is an allergy, his sensitivity could be affected by things like environmental allergens, being tired, ingesting other foods he might be sensitive to, etc. I'm not sure if those things also affect intolerances but it seems like they could. (I just assume that my littlest guy is *always* intolerant and try to avoid those foods entirely.)
For the record, I do think that teethers sometimes get loose stools from all the saliva, not that teething directly causes loose stools.
If you need to continue to avoid dairy for him, be sure that you're avoiding all forms of it until you can test his sensitivity. There are some great resources at www.foodallergy.org and www.godairyfree.org.
Best of luck!