Do artifacts appear based upon time, distance run, or something else like that? Also I've noticed I seem to get artifacts faster when there are holiday artifacts to collect. Is that also a variable? If anybody knows the spawn rate of artifacts that would be nice.
My experience is that artifacts appear based upon distance but I do not know more specifically than that. There can only be one chest grabbed per run, but if you miss the first one, other chests will continue to appear intermittently until you successfully jump to grab one, after which no more chests will appear that run.
I have also noticed that when there is an uncompleted holiday artifact quest, the rate of chests seems to increase until the quest is completed. My experience is that once the relevant holiday quest is completed, the chests revert to containing ring and mask artifacts only with no more holiday artifacts appearing.
Edit: to add some data to this, I worked out for a game that has been played entirely since the introduction of artifacts that there were a total of 78 artifacts found over 1.7 million meters, giving a rate of about 1 every 22k meters. I think powerups appear about once every 750 meters or so, so assuming that about 80% of those chests that appear are grabbed, that gives a rate of 1 in 25 powerups being a chest. That includes periods when there were holiday quests active and not active, and it includes gameplay before the Pickup Spawn was fully upgraded, so that is good to keep in mind. Since long runs can have at most one chest, that pushes the average downward. Perhaps for the initial chest it is something like 1 in 10 during the holiday quest active period, and 1 in 20 when there isn't a holiday quest active, if indeed they appear at different rates.
I have noticed that the number of chests increase when a holiday quest is available. Also, I've noticed that when a holiday quest is going on, it is more likely to get an artifact you already have.
It is difficult to get the last ring artefact because it is less likely to be in chests than other artifacts such as holiday artifacts and floral artifacts, so it's more likely that your going to get those hats before you get the other hats.
When you unlock a hat, you can equip it to your character, and you can have it equipped to more that one character, but it's more common the most of your hats or your hat is a holiday hat because, since these special quests are out for a while, but not forever like the other quests will be.