In the absence of someone more experienced commenting, here is my take:
This topic has been discussed on various climbing forums, but a strong consensus has yet to emerge in the US. (Outside the US, you are probably climbing on doubles, and so the question doesn’t arise.) I do sense an increased appreciation for climbing on doubles, and using a thin tag line does have its risks. I also suspect that a tag line made more sense when 10mm or thicker ropes were the standard compared to modern skinny ropes. Thus, for someone with less experience I would recommend using a half rope instead of a thin static line. If you are going to use a tag line, what follows is an overview of factors to consider.
First, even though you already have a single rope, consider whether your routes might be better suited for a 70m rope or for double rope technique. The first option might eliminate the need for a second rope entirely, and the second provides additional safety benefits. You’ll see more of a benefit using doubles on long multipitch routes.
A tag line, particularly a static line, is best suited for purely vertical routes (e.g. Yosemite big walls). The lack of stretch in a static line will aid in hauling; if you are climbing close to your limit, you may want to use a tag line to haul your pack. Some also prefer using a thick (10+mm and durable) single rope with a static (more durable, potentially less likely to snag) tag line in environments where the rock is particularly hard on ropes, e.g. desert sandstone. Somewhat in opposition to this are those who feel the best approach to rope-eating cracks is to ensure the second rope is suitable for leading on, so that when a rope does get caught it can still be retrieved without an effective free-solo.
Many people report lines 6mm or narrower are especially prone to tangling, kinks, and other rope management issues. (Narrow lines are also hard on your hands, whether pulling the rope after a rappel or during hauling.) One recommendation is to knot such lines at regular intervals and hang the knots on a carabiner as an alternative to flaking the line. Another is to use a stuff sack as a feeder bag rather than throwing it down and dealing with the resulting tangles. Thicker lines will help with this but reduce the weight benefit. Once you reach the 7-8mm range you might as well use a half rope; consider the 7.8mm Edelrid Apus or the 7.3mm Beale Gully.
A more wandering route presents a case where half ropes would provide a benefit in reducing rope drag. A route known for sharp edges or to be prone to rockfall presents a good argument for the insurance of a second rope. Consider also your retreat; a long series of rappels at the end of the day or due to an incoming thunderstorm will be much easier with double ropes. If you are attempting multiple rappels with odd-sized ropes, having to pull the heavier rope back up each time to feed it through the anchor means extra effort expended (refer to  below); doubles make this a non-issue.
Regarding leaving gear behind, I have difficulty imagining situations where this would be considered as anything other than an emergency option. (Obviously this does not include the occasional bit of webbing and rappel rings or “leaver biners”.) Your options for building a suitable rappel anchor halfway down a pitch may be very limited, and cams are not exactly cheap.
As to materials, Spectra and Dyneema have low melting points, rendering them vulnerable to friction and unsuitable for rappelling. They are also slicker than nylon; knots tied in them can fail at lower forces than expected (the core separates and pulls through the sheath). For this reason a triple fisherman’s bend is recommended instead of the traditional double fisherman’s bend. They are also less durable over time; repeated bending cycles can soon reduce their strength below comparable nylon alternatives.
Polypropylene floats, and that’s probably its main benefit. It’s weaker than nylon, not as durable, and also has a low melting point. The ropes you linked have a polyester sheath in order to add strength and block UV. Personally I would avoid it, especially in an environment that would abrade the sheath.
Now, some cautionary bits.
If using a static tag line, cut it a bit longer than your lead line to account for stretch (e.g. 65m rather than 60m.)
If rappelling on only a single line, use a carabiner block (tied on the thicker line) rather than assuming the knot won’t be able to pass through the anchor. (It can and has.)
If rappelling on both lines, the thinner line will feed through the rappel device faster. Feeding the thicker line through the anchor will mitigate this.
Tie stopper knots and back up your rappel.
However, canyoneers do this regularly.
Yes, and have you seen the sorts of things they call anchors?