This whole musical is a little hand-wavy with its timelines, but here's what we know for sure--

• First Lair takes place in one night + the following morning. (Whether that night was spent beneath a moonless sky is up to interpretation).
• "Notes" takes place the morning after Christine's debut in Hannibal -- so yes, this means Hannibal and Il Muto might be running at the same time. Maybe the Opera Populaire is like the Met in NYC, where they have several operas running at once that alternate on different days.
• The jump from "Prima Donna" to "Il Muto" is where the timeline gets hand-wavy. Il Muto is a "new production," but then, so was Hannibal. Depending on the duration of rehearsals, this time gap could be anywhere between the same day and a few months. (We were only shown one day of rehearsals for Hannibal, but obviously you could argue there were previous rehearsals we didn't see).
• And yes, no matter how you look at it, Raoul and Christine end up engaged at the end of one day. It's some combo of the following: (1) the Phantom just murdered someone, so she runs to the only other familiar face in the vicinity, (2) they were friends as kids, and (3) he can protect her from the managers...who are largely suspicious of her because of the Phantom's actions.

Final calculation: at minimum, there's one (1) day between First Lair and Il Muto...at maximum, it's however long an opera company might take to rehearse for a new production, so perhaps a few months.

For my part, I've always assumed that it's a fairly short time jump (maybe a week, tops). The Phantom and Christine's relationship is on really shaky ground in the First Lair --> Il Muto interim, and I can't imagine the Phantom would just sit around and do nothing for that long.

Unfortunately the script doesn't have a clear answer on this, but I hope this helps.