First of all, do you have to do the speedrun on any particular difficulty? Second, is it allowed to submit a speedrun using the remastered BioShock or does it need to be the original?

Hey the runs are not separated by difficulty with the exception of Any% hard. As for versions I'm 99% sure the remasters are in the Bioshock: The Collection page which can be found here https://www.speedrun.com/bstc#BioShock_1

Make sure to join the discord as it's pretty active and welcome!