How to install plugin without composer?
Alright. Guess the conclusion to my question is that with Craft 3 it isn't realisticly possible to use the plugin store without composer.

Luckily I finally found the problem composer had in connecting to packagist via ssl. Just in case other people have problems with composer too and are running ESET, this might help: unable-to-get-composer-running-to-install-craftcms