Are my opponents unable to cast spells for the rest of the game if I cast an Epic spell with Hive Mind on the battlefield?
Correct, Hive Mind gives every player a copy of the Epic spell. No player for whom it resolves will be able to play spells for the rest of the game.

Since Hive Mind copies the applicable spells without limitation, these copies of e.g. Enduring Ideal will have the Epic ability and no player for whom it resolves will be able to play any more spells that game.

Only the copies made by the Epic ability itself do not have the Epic ability. Therefore, you could also use any other spell or ability to copy an Epic spell and get that many copies each turn, e.g. through Chandra, the Firebrand