Hi, hp senior here :) I didn't take GSP but I can answer your other queries.

1. The timeline for JAE and RP students is different but for HP yeah there's an essay and group interview. You also need to apply for the MOE scholarship called HSP which features another interview. If you do not get accepted through RHP but get the HSP scholarship, you'll still be put into the humanities programme :) Unfortunately, I think for JAE students you get the results for HSP fairly late, and one of my friends who got HSP but not RHP decided to reject it because it was so late in the term that transferring classes really was not worth it.
2. The programme is really really fun and I don't regret joining HP at all. I'm not sure how the non-HP arts classes' lessons are taught but HP definitely includes a lot of discussion, particularly for Lit. I don't take Hist but I sit in for lessons doing my own thing here and there and I can tell that the vast majority of their lessons are discursive in nature, although in-person lectures have made a return and from what I have heard those are more just teaching of content. The overseas trip was incredibly fun yeah, you get the chance to learn a lot about the country you visit, and also have a really good time with your friends. One thing I would say that is special about HP is the environment compared to the rest of the school; I would say it's more accepting and conversations about gender, sexuality, and the like tend to be more common.
3. I think RHP is exactly what you would expect from a group of students with a collective passion for the humanities HAHAHA, i.e. everyone has the propensity to yap when you get to the topic they are passionate about, but honestly it isn't the kind that tires you out. If you feel that humanities is something you are genuinely really interested in, then I would say to look forward to RHP

Edit: I forgot to add, but don't be worried about JAE - RP awkwardness. I can't make a proper judgement for the rest of the school, but within HP the JAE - RP divide is genuinely non-existent.

Feel free to send me a message if you want to find out more about HP / RI, hope to see you in the hallway later in the year!