1. Yes, you can stay at 50 as long as you like. It was the highest level for a long time, so theres a good ammount of content to potentially dig into.

2. Maybe? Some of it very easily, some of it will require you to organize people. Kinnison is right that you wont get the same difficulty/experience that the old-timers had though.

3. This is up to you. Clear every level 50 dungeon? There are 19 of them to unlock and defeat. Defeat all the "primal threats" in their most potent form? There are like 9 of those to go through. Find out what really happened during the calamity with bahamut? Well there are 13 coils(and the harder versions of each) between you and the truth. There are 3 alliance raids you can do. You can improve relations with 5 of the beast tribes. Major side quests like Hildi and the Postmoogle are just waiting. You could work to get your zodiac weapon. Completing the sightseeing log is a true challenge. You could work to become the greatest chocobo trainer/jockey the gold saucer has ever seen. Your end game could be getting all 21 classes to level 50 and completing all their quests. Or maybe just finding the perfect outfit glamour for yourself.