1. Is it possible to stay at ARR and do endgame level 50 quests?
You can stay but very few content for ARR are added after the expansion but ARR itself has plenty of stuff you can spend time doing on until you get the expansion. Like clearing all lv50 dungeons, going through Palace of the Dead and also Beast Tribe quests.

2. Will it be possible to find party members to do said endgame activities?
Still possible depends, dungeons are still joined via roulette but as for endgame raids you might need to ask friends/fc to help out as people rarely queue them directly via Duty Finder. On top of that, you won't be able to enjoy them at the difficult level it used to have when they were first available due to max ilvl and also unsync option.

3. What would be my ultimate objective at level 50 endgame?
For ARR, glamours are your endgame if you like Fishing there's a unique quest involved with that job. Also clearing All Coil/EX Trials and finish Hildibrand's side story would be your objective.That's pretty much what you can do.

Hopefully this helps.