If you have already beaten the Invasion stages, then refer to this link on my message wall for how to install BCU for PC (which will count as a valid run). Similarly, Heavenly Tower challenges are unplayable for now, so it's recommended you obtain BCU to complete these. Jagando cannot be replayed to my knowledge because his stage in an Invasion stage, just like Idi:N's Invasion stage. In a no-Powercreep Floor 30 challenge, you cannot use units/forms/talents released after Version 6.2. In a no-Powercreep Musashi challenge, you cannot use units/forms/talents released after Version 5.9.
The term 'No powercreep' means that you cannot beat the stage with Units/Forms/Talents released after the release of the stage. This might seem hellish, torturous and psychopathic but nonetheless doable to complete all these challenges at first impression, however this is not entirely the case. If you are not lategame and wish to attempt the challenge anyways, refer to the BCU installation guides. All of these challenges are lategame so if you are not late game, I do not recommend attempting them. Idea taken from AnonymousCrouton's post and media stolen, borrowed from the Public Battle Cats Fandom server.