Good questions to start us off here!
Favorite installment? A tough question to answer, myself, since my exposure to this series hasn't been very deep until more recently. After beating a few main routes of the English release of Shin, I would've said that one, given how close it made me to the characters. That said, now that I've had a more detailed look into what lore the other games (including the original) will offer me when I eventually get around to playing them through a translator, it's hard to say where that opinion will go.
Now, if we're talking OVAs, that's easy: Episode 1 of Inyouchuu The Animation. There are plenty of fans who dismiss ITA as being too tame in comparison to the rest, but I'll go on record to argue that a tame introduction is exactly what it takes for a new fan to ease into the series. If my own first exposure to Inyouchuu had been one of the more extreme and lore-accurate scenes back in my much more naive years, I might've written it off as something far more intense than I was comfortable with, and never looked back. I'm also head-over-heels in love with the Suzuki Mirano (?) art style that it uses, to the point where I'm gradually training myself to draw my own art in that style as well.
Favorite character? Asking the real questions here! My first impulse is to say Mikoto; she's the star of the franchise, a certifiable hottie, and I've even cosplayed her. Yet, at the same time, my appreciation for Takeru has significantly grown lately as well, now that I've learned how closely her establishing story serves as an allegory to my own life and identity (see my opening post of the Introduce Yourself thread, for those who care to know).
And then there's Sui - who, personality-wise, is a lot like me... though she's much hornier.
Favorite scene? For my own emotional reasons, easily this moment right here (link to later post on this thread) from the end of Shin. But aside from that, it depends on what mood I'm in! I have to admit the Takeru+Sui interactions in Shin are absolutely adorable, with an emotionally flirty nature that prompted me to ship the two of them in my fanfic. As far as OVA stuff, the famed Nightmare x Mikoto scene definitely ranks highly, although there are plenty of good moments across most of them. And if you want my favorite CG... that's a tough one. Maybe I'll come back to it after I've actually played a few more of them in context.
Favorite songs? Oh, gosh. TrioDesign cooked up some absolute masterpieces in these games, and it's hard to even narrow down my favorites. I'm familiar with the soundtracks from the first two games and Shin. From that, I can say all three OPs are amazing (No Way, Feeling This, and Undermine), as are the sword/battle songs from each. The recurring song used in particular at the beginning of Takeru's route in Shin is immensely catchy. Shoku has a few variations on a common leitmotif for general story music that I also enjoy quite a bit. There's probably more I'd call out, too, but those are some of the big ones for me.
I just wish I knew actual titles of most of those. There was a "Best Of" CD including a mix of music from the first two games, which actually specifies song names for the BGM tracks, but I don't know the rest of them outside of what I "headcanon" if you will.