These are perfectly reasonable questions, without completely straightforward answers.
Our best evidence about such things takes the form of photographs, with paintings and drawings (by artists who knew what they were doing) a good second-best. Preserved ships and modern full-sized replicas also are useful in establishing what actually works and what doesn't.
In theory, all the shrouds have to be in nearly-identical tension or they can't do their job. That means that on the weather side of the ship they're extremely taut, and on the lee side (in a strong wind) they're slack. In just about every photograph of a sailing ship I've ever seen, the upper deadeyes are lined up pretty precisely in a straight line parallel to the lower ones. That's also how they've been arranged in every real ship and replica I've ever seen.
Most of the preserved ships, and all of the photographs, in question date from the eighteenth and nineteenth centuries. During that period the process of lining up the deadeyes was made easier by a fitting called a "sheer pole" - a simple iron or wood rod that was seized to the shrouds just above the upper deadeyes. The sheer pole made sure that all the deadeyes were in line - even if some of the shrouds were a little more taut than others.
The sheer pole apparently made its appearance late in the seventeenth century. For earlier periods, all we have to consult are paintings and drawings. The vast majority of the ones I've seen show the deadeyes neatly lined up - sheer pole or no. A couple of weeks ago my students and I took a field trip to Jamestown Settlement, and I took some pictures of the replicas of the early-seventeenth-century ships Susan Constant, Godspeed, and Discovery. Their deadeyes are lined up quite precisely - and they don't have sheer poles. The guys who work on them tighten the shrouds every few days (more frequently when the ships are at sea). I'm pretty sure the rope they use is some sort of non-stretchy modern synthetic (though it looks pretty convincing), but the appearance of those ships certainly matches what little contemporary graphic evidence we have.
I just looked through the contemporary pictures in a new book, The Tudor Navy, by Arthur Nelson. In a lot of those old prints and paintings it's hard to see the details of the rigging, but in those where the deadeyes are visible they're lined up in parallel rows.
Remember that the seizings and lanyards aren't permanent, and can be adjusted fairly easily. It was common practice to slack off the lanyard of the foremost shroud on the lee side of the ship, to give the lower yard a little more room to swing. And I once watched a sailor on board the replica schooner Bluenose II while he overhauled the lower rigging. He loosed the ends of the lanyards, then untied the seizings of each lower shroud, pulled the shroud a little further around the deadeye, replaced the seizings, and hove the lanyard taut. When he was finished, the deadeyes were all arranged in a nice, straight row - slightly closer to the lower deadeyes than they'd been before he started.
With all that said, I do have my doubts about how neatly such things were actually done in practice. I suspect that a ship at sea, far out of sight of land, that had just been through a storm, wasn't quite as ship-shape as the paintings and photos lead us to believe - with respect to the alignment of the deadeyes and plenty of other stuff.
I'd also emphasize again that we're talking here about practices that became common long after 1492. Some good reconstructions of Columbus's ships don't have deadeyes; their shrouds are set up with block and tackle. And the few contemporary pictures from the period suggest that ships' rigging in the late fifteenth century was in general treated a good bit more casually than it was later.