If you're installing a package via the package manager, it will very quickly tell you if you can install it on your specific version of python. Unless you're downloading some rather obscure and un-loved library where the author didn't explicitly state which versions of python they support.
Multiple package managers: There has only been 2 big ones from my long-term general usage of python. Easy-install and pip, the former of which is falling in favor but still semi-supported. Pretty much everything runs off of pip. What may be confusing you, and does confuse me at times as well, is their naming and the installation instructions as provided by library authors. E.g. some say python setup.py -install others just tell you to "pip install" it. Some would say use "setuptools", etc. Other would tell you to use things such as "conda" or "anaconda", pipx, and to create virtualenvs. All secondary, but things that should not ideally distract you from just plain using pip.
3. This has also been getting a whole lot better in the last 5 or so years. Microsoft has been funding dev-time to make the ecosystem for python (including extension compilation) much more pleasant in the Windows space. Also, the package managers and library authors are doing a whole lot better in that binary distributions are much more prominent so the compilation of the extensions never has to happen on your machine.