I think that 5mm cord is fine for abseils, even when heavily loaded as its strength is 5.5 kN when unknotted, and a little less when tied with an overhand knot (same as Euro Death Knot), but if you want to feel super safe, then 6mm may be best (see this vid of a retreat of the Eiger using 6mm threads due to loading). I prefer cord over skinny tape, as flat tape becomes very weak under even the slightest of abrasion, and is much less cut-resistant than cord (it also rips when loaded, which in my experience leads to catastrophic failure long before a length of 5mm cord would break).
I find that ‘chaining’ rap cord works well, as it means it stays untangled better (to do this tie an overhand knot in end, then feed the cord through this to form a small loop, then feed the cord through this loop, and repeat, creating a series of loops that untie when pulled from one end). When rapping always clip one end of the cord into something, even when storing in a pocket, as it’s very easy to lose (it will snake out of pocket faster than your money in the kid’s summer holidays).
If you don’t have a knife you can cut cord with an adze or hammer, or bash with a rock (use the edge of a nut tool, or even the edge of a nut itself, and bash this), or just run it up and down the rock to cut.
BTW If you come across any fixed cord always check the knot (a figure of 8 tied instead of a EDK is deadly on thin cord), and the cord itself, and if you can cut it away and use fresh cord, as thin rap material is only really strong enough to be used with total certainty when new. Saying that cord is probably far more UV resistant than tape, which quickly gets stiff and weak, whereas the sheath of the cord tends to shield the core.
Hope that helps.