Every character I've started that did not go to Whiterun and kill the dragon that attacks there has not encountered any dragons. Do they only start appearing after you've killed the dragon at Whiterun? Does anyone know?

They don't appear until after you've killed the one at the Western Watchtower.

You should probably have put this in spoliers. And yes.

I believe there's sufficient foreshadowing in the first 20minutes of Skyrim to consider the Whiterun Dragon fight not a spoiler... At least no more a spoiler than saying

Well, the fact that Dragons will show up during the MQ is not a Spoiler. But, the details about how and when they show up is a spoiler.

I will move this now.

Spoiler? Really? I don't agree, but thats fine. Thanks for moving it.

Also, the random attacks don't really start until after you go to that place with that person.
So if you don't start the MQ: are there no dragons at the word walls? What about the Dragon Priests?

Damn, it! You ruined my game - I've never died.

The random encounters start after you complete the quest to kill the dragon at the Whiterun Western Watchtower. I always play right up to that point and do nothing else just so the dragons spawn.

Also, until that point, dragons' lairs are populated by some other type of enemy. Witches, bandits, wild animals, etc. Once the dragon moves in, you can usually find remnants (or charred/burnt remains or maybe some bones) of what used to live there.

You are correct that the dragons appear at the dragon wall lairs once the Whiterun Watchtower quest event concludes, but I don't think random spawns will occur until the Kynesgrove incident with Delphine, UNLESS you've began the civil war quests. One of the plot archs in this questline involve a truce so that all efforts can be diverted to fighting dragons, so random spawns on military quest locations do take place. However once the civil war is ended, and if you haven't opted for the truce settlement, no random spawns occur again until Kynesgrove

So beginning a game play-through as a civil war participant is a nice way to build up some dragon souls ahead of time, then you can go out and see the world without a random dragon cluttering your adventure every 30 -60 minutes. As far as activating shouts go , the supply of specific locale related dragon kills is sufficient to activate most of the best ones, but you shouldn't count on activating all of them. The only drawback, and its a liveable one, is that you can't do the Hrothgar mountain top training with Parth. (No "Clear Skies" either) or launch the "Repair The White Phial" quest.

I can speak with 100% certainty that the random dragon spawns occur after killing the dragon at Whiterun Watchtower. I haven't seen spoken to Tullius or Ulfric yet in my current save, and I've run into random dragon spawns. I didn't do Kynesgrove resurrection sequence until about three days ago, and I still got random spawns up to that quest.

That's what those remains are. It never occurred to me.

This has been my experience on my 2nd character. I had 2 random attacks early on after the Whiterun Watchtower incident around LVL 11-12 and then no more randoms (I killed a few at the word walls though) until after like LVL 52 when I did A Blade in the Dark.

Thaaat's pretty odd then, since I'm beyond certain that I've gotten random dragon attacks on all three of my characters way before completing A Blade in the Dark. On my current character, I've had a couple dragons outside Riften, near Honningbrew Meadery, a couple on the Solitude Docks, this one just appeared out of nowhere when I existed a Draugr lair in Hjaalmarch, the list goes on.

Gallarialchiban is correct. If you would like to "force" random dragon spawns after you've downed the one at the western watchtower, simply rest for a minimum of 5 in game days. Next time you go wandering outside one will spawn. Sometimes they attack, sometimes they don't, but they do spawn.

Oh, I believe you and I'm sure most other people do, too. This game is one in which 100 people can buy the same game and get 200 different experiences with any one thing.

Yes , my experience as well. Originally my goal was to leave all of the mainquest out of the picture, and be a civil war hero instead, but you HAVE to do the Watchtower , before Whiteruns Jarl Balgruff will do his part as linchpin for either side of the quest. Once the civil war is done, no random dragons from level 14 -53 thus far. I find it much more exciting to discover dragon lairs, Artic Point in particular was a cool experience, than just have one thrown in your face every hour.

I think that's the one I came across yesterday. Kind of a camp with Aura Whisper at the Word Wall?

Dragons seems to take a brief pause during the "The Horn of Jurgen Windcaller". They do reappear randomly after that specific quest during the MQ.