At a place like Lao Sze Chuan, they indicate on the menu if a dish is spicy by using one, two or three peppers. Not everyone at LSC is fluent in English, but I'm sure they are good enough to understand to moderate the heat of a dish. Of course, if you are not a fan of heat, be careful with LSC's cabbage, which is served to start the meal.
I doubt you'll encounter much difficulty with language at most Thai and Indian places, including Spoon Thai. You can just ask if a dish is spicy, and you can ask them to moderate the heat of a dish and for those that want more heat, ask for hot peppers on the side. You'll probably encounter considerably more spice at a Thai restaurant than an Indian restaurant.
In short, I'm sure you'll be easily able to communicate your desire for minimal spice everywhere you go.