Will I be playing with other participants?

Its more fun with other participants. I doubt it would be enjoyable alone.

There is an option to pay for a private room which essentially is purchasing all the spots in the room. If you just book the seats, it leaves spots open for others to book. No guarantee others will or won't join you though.

You have the option to pay for a private room for a much higher price, but you will book with the possibility of other people booking the same time. If it helps, most of the time I booked, it was just my group.