I have to confess I've never seen rigging line referred to as "Size 9" or "Size 1." I'm assuming that what's being referred to is simply the circumference of the line - in inches.
If you've got the instruction book for the Bluejacket kit on the way, you'll probably find it answers just about every practical question you may have about the rigging of this particular ship - to the extent that such questions can be answered. Despite the fame of the Constitution, and the amount of research about her that's been done over the years, there's still plenty of room for guesswork, inference, and interpretation about many details.
The most important thing to worry about regarding "types" of rope, in modeler's terms, is the direction of the "lay." "Cable-laid" rope is spun up left-handed. (If you look at a strand of it running vertically, the strands go up and to the left.) "Hawser-laid" rope is right-handed. Throughout the period of the sailing warship, nearly all running rigging lines were hawser-laid. In the British Royal Navy (and probably in the U.S. Navy), cable-laid rope was reserved for major lines of the standing rigging, such as the stays and shrouds. (Typically, each of the three strands was hawser-laid, and the three were spun together left-handed. That system was believed to give great strength - though how much difference the lay of the rope actually made is questionable.)
I suspect that reference to "nine-stranded rope" actually refers to three strands, each spun up from three smaller strands. That's actually how most large-diameter rope was made.
Shrouds were generally made of "shroud-laid" rope, which had four strands rather than the usual three. That's really a matter of academic interest, though; in practical, modeling terms the eye can't detect how many strands are in a rope unless the end is visible (and unlaid). In my little model of the frigate Hancock I made some of the lighter lines with two strands (which is utterly inaccurate), but nobody's ever noticed - and I'd have trouble at this point identifying the two-stranded lines myself.
In those photos of the Hull model it's hard to tell (at least on my little monitor screen) which lines are cable-laid and which are hawser-laid. Most of the shrouds and stays seem to be hawser-laid, but I think I can see a few cable-laid lines running across a couple of the pictures. That is not, however, definitive evidence of how the real ship was rigged. The person (or people) who built this model quite obviously had to work with a severely limited range of materials There's a remarkable contrast between the workmanship of the hull and deck fittings of that model, on one hand, and the rigging, on the other. The guns are downright primitive, such things as the capstan and fiferails aren't much better, and the modeler apparently was completely defeated by the steering wheel. The rigging, on the other hand, is remarkably thorough and detailed - obviously the work of somebody who knew exactly what the real thing looked like. But small-diameter cable-laid line probably was hard to find (as it is today). Or maybe the real shrouds were hawser-laid. I don't think anybody knows for absolute certain.
As I've remarked in earlier threads, I really wish some scholar would undertake a thorough study of this model and write an article or monograph about it. It may well be the most historically important ship model in the U.S., but I have the impression that relatively little is known for certain about it. I wonder, for example, just how much restoration and repair it's undergone over the centuries. Is the rigging all original, or has some of it been replaced at some time or other? (In one of those pictures some shiny brass belaying pins are visible; they look remarkably like they were bought fairly recently from a ship model supply company.)
For a first attempt at rigging a ship model, I frankly don't recommend getting obsessed with the mechanics of rope making. Concentrate on "learning the ropes" - understanding what each of the rigging lines is for, and how they all work together to make the ship function. There are various sources of model rope; my own preference for many years has been silk thread, but it's been so long since I've bought any that I really wouldn't know where to find it. (I've seen some ads for it on the web, but I've never tried any of those sources.) Model Expo sells a "cotton-poly mix" that I rather like; it has a good, rope-like lay and reasonable colors. Bluejacket sells several sizes of linen, a traditional material that some modelers really like; I do too, though I haven't had much success with it on the small scales to which I tend to gravitate. (It also only comes in black and white, so running rigging line has to be dyed.) I'm sure there are some good sources for rigging thread in Europe, but I'm not familiar with them.
Virtually all commercially-available thread seems to be spun right-handed (i.e., cable-laid). Modelers with some experience often end up making their own "rope-making machines," which can be used to spin fine thread up in either direction. (Model Expo offers a kit to build a rope-making machine. I've only seen pictures of it; it differs from the traditional design in that it doesn't have a "top" to set the "pitch" of the twist. It may work perfectly well, but I have reservations about it. For the time being I'll stick with my own, primitive version.) I don't recommend worrying too much about this sort of thing on one's first rigging project, though.
Hope all that helps a little. Good luck.