If Strawberry Perl is installed from their .exe file, how can we find out which modules are installed

The core modules (ie the modules that are installed by default in all builds of Windows perl) will be in the perl/lib folder, and all additional modules will be in perl/vendor/lib.

There's also DISTRIBUTIONS.txt that lists all of the modules that were installed into perl/vendor/lib.
Look for it in the top level perl folder. (I assume the executable file installs DISTRIBUTIONS.txt - though I always install Strawberry's "portable edition" that comes as a zip file, so I can't be sure.)

Other than that, you can always check for the existence of a specific module by trying to load it.

C:\>perl -MSome::Module -e 1
Can't locate Some/Module.pm in @INC (you may need to install the Some:
+:Module module) (@INC contains: C:/_64/strawberry-5.28.0/perl/site/li
+b C:/_64/strawberry-5.28.0/perl/vendor/lib C:/_64/strawberry-5.28.0/p
+erl/lib).
BEGIN failed--compilation aborted.

C:\>perl -MLWP::UserAgent -e 1

C:\>
[download]

Cheers,
Rob