On several meters you can actually leave the date off, this however should be limited to pre addressed return envelopes. These wouldn't need to be placed in a special box. There are thousands of cities without a special drop box for metered mail. So, I don't think it is a requirement, but metered mail is usually faster so if you're concerned about your days outstanding on your invoices I would use a metered mail drop box if it is available.
To answer your second question if there is a date it does expire. The reason for this is first class mail is time sensitive, you might notice that most of the metered mail you recieve is not cancelled, the dated metered mail cancels itself. If you are sending out standard mail, the postmaster probably won't care if it is a day or two late, but it is his call.