I'm pretty far into P4G at this point (mid-November) and it is the only Persona game or any SMT game I have played. So keep that in mind for my responses to the OP's points.
1) I would prefer a better relationship system. I am dating Chie and have managed to keep all other relationships platonic, but ever since maxing out the social link, I feel like I've only had one opportunity to spend time with her. I'm not looking for any kind of sex scenes or anything like that, but still, if she's my girlfriend, we should go on dates. I also find it weird and kind of stupid that the relationship seems to be secret from the rest of the group. Sure, I could see in real life how it could negatively impact your social links with the rest of the girls, but this is just a game and does not have to go that far.
2) I have no problem with the music except for the repetitiveness. I wouldn't mind a bit of a change in music from time to time. I admit, I don't understand much of the lyrics, but then again I'm not really trying to. The Engrish does not bother me at all, compared to hearing the same three songs over and over and over.
3) I don't feel like I've really had to grind at all. All my characters are levelled into the 60's with a couple in the 70's. I keep changing up the team regularly, usually keeping one "battle pair" together for the bonus attack they can provide. Going back through every level to defeat the powerful shadow that takes residence (which is nearly always harder than the shadow persona originally present) and going back to retrieve items for Quests is all I've done as far as grinding, and I'm not having much difficulty at all, except for the occasional boss.
4) The dungeons could definitely be more interesting. Not sure how though. I'm trying to recall what other RPG's have done to make their dungeons interesting, but I'm drawing a blank. I have no problem with the random layouts, but that's probably from playing too much Diablo 2 back in the day.
5) Totally agree with this. I think my wife would really enjoy P4, even if playing as a guy, but I know she would prefer to play as a girl. I'm sure this is true for many female gamers, and while it would involve quite a bit of writing to change the character dynamics depending on the protagonist's gender, I think the effort would really pay off for the player. Admittedly, I don't think every character should have the option of being in a relationship with the protagonist regardless of sex, but there should be the option for at least one or more. I think I would just find it too unbelievable if I played as a girl yet could date any female character and same if I was a guy with the guy characters. But one or two of each being willing I would find believable.
6) Since I have been playing on my Vita, this is not an issue for me, but I can definitely see how it would be convenient to have on a PS2. Even if it was just a save at the start of each floor of a dungeon. For me, though, I can just put the Vita to sleep and go on my way without needing to worry about it. I don't have one, but the 3DS is capable of this as well. Depending on what systems P5 comes to, a quicksave may not be needed at all. Additionally, the PS4 is supposed to have this kind of functionality, so if you need to leave, just put the system to sleep and pick up right where you left off when you return.
I'm going to have to echo Eric Neu above me though, the Investigation period is nothing but frustrating. Having to wait for a particular day in order to meet the right person you need to talk to in order to get information about who to talk to next is annoying! I feel like I have wasted weeks of game time because I couldn't go in the TV without more information, no one was interested in just hanging out to let me upgrade my social links, and there was just nothing to do. After each day verifying that the person I needed to speak to wasn't available, I'd just spend the afternoon time studying in the library or similar so at least something constructive happened. I liked the rescues that I could take care of right after the person went missing, since I always felt a sense of urgency and wanted to get them out as fast as possible.