Is a competition to memorize Prophetic hadiths, by writing and correcting them via WhatsApp, that involves no prizes or wagers, considered a mockery of hadith, or does it contain any aspect of impermissibility?

What you are doing by studying Islamic knowledge is rewarded if you purify your intention. The mistake of one of you in a hadith is not a mockery of it. It is feared that your question is from the whispers of Satan that might turn you away from good.