They're generally called "box trailers", here, of which a "horse trailer" is a special case.
Trash cans are all over the place. Most homeowners have 30-gallon but trash collection companies often give their customers 45-gallon cans on wheels. They're standard sized, matching lifts on the trucks so the pickup is semi-automatic.
Even every-other-week would get smelly. Ours is once per week and even that gets pretty ripe. We don't have recycle pickup (it's available but at higher cost - I think). No need. Everything fits in one 45-gallon trash barrel per week.
Open or not. Yes, a personal "truck" here usually means a pickup, though not always open back. Caps and covers are quite popular. They're often four, five, or six seats, too.
I don't think insurance is available by the day here. Comprehensive must mean something else to you than it does us, too. "Comprehensive" means non-accident loss (like a tree falling on it), here. It covers the value of the vehicle only. "Liability" covers damage caused by the vehicle and that's what's required in most states, here. "Comprehensive" and "Collision" insurance coverage isn't required by law, though it may be contractually required for a loan.