There were various ways to rig a lateen yard. The only one I've seen in operation is the mizzen yard of the replica of the Susan Constant, at Jamestown.
That yard is rigged (quite accurately, I think) with a fairly simple "peak halyard" tackle running from the upper end of the yard to a block on the crosstrees, and then to a belaying point on the deck. The sail is rigged with three or four "brails," which run from grommets stitched into the foot of the sail to blocks on the yard, and then to the deck. Two tackles called (oddly enough) "bowlines" run from the lower end of the yard to blocks secured to the aftermost main shroud on each side; they're used to swing the yard back and forth. The other major piece of rigging is the sheet, which runs from the lower aft corner of the sail to the deck.
A couple of times when my students and I have been in Jamestown on field trips, a staff member has taught four or five of the students how to handle the mizzen sail. The first step is to slack off the tacks and the sheet. Then the brails are hauled up, turning the sail into a loose bundle on the yard. The peak halyard is then hauled taut, bringing the yard vertical next to the mast. Then one or two people grab the bottom end of the yard and walk it around the after side of the mast to the other side. The peak halyard and the brails are slacked off. The sheet and bowlines are hauled taut, and now the sail is ready to handle on the new tack.
The Susan Constant has a mizzen stay; it doesn't interfere with the process of taking in and resetting the mizzen because the yard is walked around the after side of the mast. Whether Columbus's ships had mizzen stays I don't know - and I don't think anybody else knows for sure either. But there was no practical reason why they couldn't.
I once asked the "captain" of the Susan Constant how long that evolution took (wiith with a crew of genuine sailors instead of landlubbing college students). He said two or three guys could do it in one or two minutes. (Obviously it would take more crew in a bigger ship, but I don't imagine the time would change much.) He also said that when the Susan Constant was out on a routine sail in the James River and Hampton Roads, with lots of tacking to be done, he simply didn't bother to set the mizzen. He said the ship handles ok without it, as long as the spritsail isn't set. (I have a sneaking suspicion that the carefully concealed diesel engine helps a bit.)
Hiding the ends of a seizing gets harder as the line gets smaller. The usual full-size practice (as usual, it's difficult to describe ropework verbally) is to start the process by forming a long, skinny bight (unsecured loop) of the seizing line and hold it with one hand alongside the rope or spar being seized. Wind the seizing line around the rope or spar, taking it around the bight. When you have enough turns, pass the end of the seizing line through the bight. Pull both ends taut, so the bight gets yanked under the seizing (and clamped by it to the rope or spar being served). Snip off the ends and tuck them under the seizing.
On a moderate-sized ship model that method works pretty well if you're seizing a relatively large rope or wrapping the line around a relatively large spar (e.g., the wooldings on a mast). A touch of white glue or CA will hold everything in place. For smaller applications, in which the line is thin, slippery, and droopy, I usually just pass each end under a couple of turns of the serving, pull them taut, apply a tiny drop of white glue, let it dry, and snip off the ends of the serving line as close as I can (frequently resorting to a razor blade to do it).
Hope that hellps a little.