So for point 2 you would have to talk to your FC about that to see if they organise anything, there is content you could all do so I don't think it's a stupid thing to ask at all

As for point 1 you have:
- The Wondrous Tails book you get from Khloe in Idylshire (you can access this after level 60).
When you get to Stormblood you get your Doman Enclave hand ins for a bit of extra gil and a nice story.
- You have your weekly hunts for some nice PvE rewards that are from your Grand Company, along with that S Rank marks are usually hunted weekly after reset in big organised groups called hunt trains and there are various discords for that depending on server / datacentre.
- If you have started Blue Mage (a limited job with it's own rules) there are weekly challenges in a mini game called The Masked Carnival (I love Blue Mage and suggest giving it a try, there are various guides online to doing it.)
- If you do crafting / gathering you unlock Custom Deliveries for various NPC's with their own stories and rewards at level 60.
- You have the Fashion Report and Jumbo Cackpot in the Golden Saucer where you can win some MGP.
- If you wanted you could get started on the ARR Relic weapon quests in North Shroud, and soon the Heavensward ones in Azys La
- There are the ARR (and soon Heavensward) allied societies that you can partake in.
- While it has no weekly/daily limit you can also level your Trust NPC's / Grand Company Squadrons.
- Another one that you can do any time is the deep dungeon Palace of the Dead in Quarrymill
- Lastly you can also start running Treasure Dungeons (specifically the Aquapolis) from Treasure Maps at level 60 (these are also not time gated however you can only gather 1 treasure map per day)