Donna has covered the bases so I'll just echo her advice:
Yes, coopculture is the official ticketing site.
Yes, it you purchase an underground tour through the coopculture site, you must also purchase a general-entry ticket.
No, there's no indication yet if they'll be offering undergournd tour bookings beyond June so you need to watch the site for updates. If they follow the pattern of the last 6 months, tickets will become available on the third Monday of each month, at 9:00 AM ROME TIME, for the following month, and they sell out very quickly.
No, there's no time when the Colosseum is not crowded. As she said, the Colosseum and Vatican Museums are the two most-visited attractions in Rome so unless it's dead of winter, maybe, they're going to be very, very busy.
It doesn't really matter in what order you do the Forum and Colosseum in BUT your ticket also includes the Palatine - accessed from the Forum, and the two considered a single attraction - and I highly recommend seeing that as well as it's very interesting! As she recommended, tour those first to avoid the worst of summer sun/heat.