Ah, well World has Tailrider Safari, but that's only 3 Palicos (which is just one group). And Palico level up when they are on hunt with you, but there aren't any training or anything.
And yeah, I just checked, you can go inside the camp and change equipment. I didn't actually check if you can only change armour or change weapon too, but there's a button for change equipment.
I finally defeated Pink Rathian! Just captured it. Upgraded my weapon to Thunder one (Pulsar Shotel III), and finally got it. Now I think it's time to hunt Nergigante, but I am going to upgrade my armour first. Thinking about farming Rathian (normal one, not Pink). Much better than Bone, and I already have most of the stuff, just need some more.
Sorry to hear about your JoyCon. I also got my Switch at launch, and it has seen some rough use, but fortunately everything works fine. Though when using docked I prefer Pro Controller, it's really good. Maybe give it a try?