One question I have is, if your server check showed several red check marks AND you have a lower version of PHP than required, then how did you get Moodle installed in the first place? It should not have installed with those kind of errors in the server environment in the first place.
Usually a warning which will allow the installation to continue will be orange and advise checking, while a failed test would be red and should prevent installation - you've said the two tests you mentioned were red checkmarks, while a lower PHP version should also have given a red failed check.
Yes that 0.0.2 would make a difference. If Moodle says 5.6.5 is the minimum requirement then that's what you need.
Please upgrade your server environment - make sure EVERYTHING is installed the way it should be, with no red check marks, minimum versions of server side software - PHP, MySQL, etc. THEN see if Boost and everything else works properly. The chances are at the moment Boost is just the first thing you have come across and there will be other things that dont work properly. They may be in the background and less visible, but they will make your site unstable, if not entirely broken. That after all is why there are minimum requirements and server environment checks.