Well, I'd imagine that either the dragons would have new technological advancements added as a type of armor to them or...
OR.
If Pern did progress into what is Earth today, there would be really no need for the dragons. New technology would be a better protection against Thread and such. Even wars would have little use for dragons what with all the nuclear bombs that could just be sent everywhere. The dragons might just become a past time rather than a real honor and perhaps they would be more like docile housepets. Though I guess they could be useful in wars, wreaking havoc on people not properly protected. Though I don't know if even the hide of a dragon could stand against some of the weapons the military has created now.
Up until a certain point though, dragon transportation would still be much greater then other technologies. You could never beat them in speed but they are more limited in capacity. Still, after a certain point human population proceeding the same it would be harder to feed and house them and then numbers would go down that way.
On the other hand, if dragons were prioritized then as tech appeared it would be applied in ways that do not compete as much with dragons.
I don't think Pern has the potential to become as industrialized as Earth. Several reasons.
The "Parallels Earth, Resources Negligible" short story implies that there really isn't enough on Pern to make it into Earth. From the presence of Thread all the native fauna is relatively small (ie: no dinosaurs, large carbon based life forms) and therefore NO OIL. Not to mention the extreme lack of metals. No fuel, no plastic, and no copious amounts of mechanized equipment on a large scale.
So they'd possibly progress somehow in tech, but it wouldn't be like Earth at all.
No idea... but I know you can make ethanol from corn, which can be used to fuel vehicles, however, it takes HUGE fields to make enough to run just a few small cars... And even then, it has to be mixed in with fossil fuels in order to be truly effective.
It'd be hard to trade dragons... Unless you were trading the Riders too... I think they would be considered sacred, perhaps even worshipped if the world were to develop with them... Maybe... But there are all kinds of possibilities.
It's possible that Riders would be hired by the extremely wealthy as private novelties and as a replacement for private jets.
The fact that dragons can teleport is extremely, extremely useful. I could see it used potentially in medical transportation (if they can keep eggs from dying between why not a heart?). Part of the limit in heart transplants is distance to transport. Dragons can be anywhere in 8 seconds.
It would have that great advantage... Despite not carrying as much, their teleportation abilities and lack of need for a landing strip would make them much more ideal for rescue operations and such...
Well, for one thing, Pern is extremely malleable. It can be any place you want it to be really, between different fandom Weyrs. It has just enough structure to start somewhere, and the dragon match system, well, I get a certain sense of wild satisfaction from that. Lots of other bonded creature RPs that I've seen don't have a systematic way of it and it just feels like the people with dragons-etc. just have them because the plot said so.