Learn French: Nous avons installé le bivouac sous la pluie.

In French, you are considered to be "under" the rain rather than "in" it. Sous la pluie is the standard, fixed expression for "in the rain." You will see this preposition used with other weather conditions too, such as sous le soleil (in the sun).

Since this lesson is about pronominal verbs, how does installer change meaning if we use s'installer?

The non-pronominal installer used here means "to set up" or "to install" an object (the camp). If you use the pronominal s'installer, the meaning shifts to "to settle in," "to make oneself comfortable," or "to move in." For example, Nous nous sommes installés dans la tente would mean "We settled into the tent."

Is bivouac a common word in French?

Yes, it is much more common in everyday French than its English equivalent. While "bivouac" in English can sound military or highly technical, French hikers and outdoors enthusiasts use le bivouac routinely to talk about "wild camping" or setting up a temporary camp for the night.

Could another verb be used instead of installer to mean "set up" in this context?

Absolutely. Monter is very commonly used for assembling or setting up outdoor gear. You will frequently hear monter le camp (to set up camp) or monter la tente (to pitch the tent). Both work perfectly here.