While walking down the street on a hot Shabbat afternoon, is it permissable to walk through a gentile’s lawn sprinkler to cool off a bit (and increase my Oneg Shabbat)? I’m not asking about getting soaked as in a shower.

There is no clear prohibition. I can think of the following issues. 1. Soaking one's clothes to the extent that if squeezed water will come out is prohibited. In this case there seems to be no intention to do so, and it won't clearly happen. 2. Trespassing - if the water is hitting the sidewalk, no problem. Or if the gentile gives permission. 3. Even if there is no eruv, there would be no problem of walking around with the water unless you actually catch some in your palm or a glass. 4. Redirecting the water to the grass might consttute watering the lawn, but there is certainly no intention, nor would you care. It seems difficult to say this is an issue. 5. Although there is no technical issue, I would be concerned that walking through sprinklers on Shabbat might become a (bad) habit.