I am currently practising running Lost Legacy on crushing difficulty. Why is there not a catagory for it at the moment? Are crushing runs allowed? Would appreciate any responses, especially from the mods. Thank you.

I don't think anyone has shown interest in running crushing before. No reason to add the category if no-one wants to run it.

By all means you can run crushing, but there wont be a category added for it specifically.

I would like to stream the run to my youtube channel for evidence-sake. If there is no catagory for it on Speedrun.com do I have the right to claim that my run is, potentially, the world record run? Provided that the run would actually be the fastest.

This leaderboard is only the "official" one because we all agree it is. If your run is the fastest recorded run on crushing, then it is crushing world record. Regardless whether there's a category or not.

There is now a category: https://www.speedrun.com/ucce/full_game#Crushing_Glitchless