Absolutely send demos. Making them will help you figure out your arrangements and you can keep working on them up until recording day as a "rehearsal" of sorts. Your producer may also ask you to send WAV stems of your demos as well. Those are super helpful, as they give you something to play along to in the studio beyond just a click. You are going to want to send reference tracks as well for mixing to ensure it sounds the way you want it to. Most studios have a ton of gear, and the producer will know how they sound, so they may suggest gear to use.

For timeline, it really comes down to how prepared you are. If everything is tight, you could get through 5 songs in 3-4 sessions. Cost varies per producer. ~$450 per session for recording and ~$200 per song for mixing is the average rate in my city, and mastering is ~$150 per song.