This is something I considered a long time ago. Fighting against a flighty, mobile enemy such as a dragon, standard weapons training (Kunst das fetchens et al.) would be considerably less useful.
On foot, these knights would have to be rather athletic and acrobatic. Attacking smaller flighty dragons would likely be the job of crossbowmen or longbowmen~ puncture its wings more than a couple of times and it will have trouble staying aloft.
For the larger ground based ones, one assumes they get the dragon-standard: adamantine scales; so bolts become less effective. Instead, our athletic knights probably master things like pole-vaulting onto its back and then using tapered longswords or stilletos to penetrate the gaps between scales, whilst other parts of the division keep it distracted.
This requires us to know how deadly dragonfire is, of course. If its 9 billion degrees or other such madness, distraction is merely going to be blowing raspberries in front of the beast and running like the clappers.
If its less ridiculous than that; specialised tower shields with ground spikes might work, even if they are disposable.
In general, these knights are not in full plate or even chainmail. A helm and a breastplate are probably all they have. No point having armour if a single tail swipe drops you on your ass helpless and ready to be clawed in the gut. They rely primarily on dodging.
So overall, reflexes, perception, instinct. Courage can be militaristically drilled in if needed, but would also be needed. Strength and athleticism enough to do all that crazy. As you say, this is a lifelong career, not something a standard mercenary drops into very easily (though vice-versa is plausible).
As a result, we assume there is special dragoon squad in every batallion of knights; this amount of training is probably too expensive logistically to give to the entire army, though those might get a 'Dragons 101' course if they're common opponents.
EDIT: To consider your highlights
Dragon Defence~ Assuming you're talking about regular troops and we don't have magic; it would be difficult at the best of times. A keep away policy is likely best- long polearms and crossbows again. Again, this is all dependent on the range and potency of dragonsbreath. Dragons 101 may also cover the ability to rapidly disperse formation in an orderly fashion. This is typically suicidal on a regular battlefield, but here it would be advantageous to get out of phalanx and beat the creatures from as many angles of attack as possible. Although this assumes that, regardless of its general potency, it can still be zerg-rushed to an extent. If its good at blowing flankers away with its wings and tail, keep away is the only thing you have, I'm afraid.
Fighting from the back: Long lances or poleaxes. Assuming the dragon also has a weight limit, we don't want our riders in heavy armour again. The training for this is mostly going to be akin to light cavalry training: rideby attacks and disruption, something our dragoneers will excel in. After all whats more disruptive in a field battle than something that forces you to break formation?
Props for training Probably an assault course. Depending on the level of mechanics in the setting, it should have as many unpredictable moving sections as possible. Early training menus may start with lesser beasts (bull-fighting etc.) to get people used to it. Attemping to ride wild horses is another good start. Or, if dragon-taming is at such a level where a normally docile beast can get aggressive on command for the purposes of training, then that is the probably the best of all. For this, they could use fake polearms and plant flags or something rather than stabbing it obviously.
EDIT 2: Didn't notice castle design as a parameter before. Did I miss it?
Anyway~ Castle Design: First and foremost, eliminating 'firing lanes' is critical. You don't want the dragon to be able to fly straight over your courtyard. Hence, you place towers in unsymmetrical locations so if a dragon wants to fly over your castle and blast with fire, its gotta take a really curvy route. Secondly, you add diagonally upwards facing spikes to your crenulated walls, so a fly by attack against guys on the battlements is more difficult. Thirdly, you add in murder strips, so if our silly dragon does fly the curvy route, it gets to a point in the defence where it gets barraged from shots from both sides.
Again, all of this depends on the range of dragonfire. If our dragon can sit half a mile in the air, out of arrow reach, and lay some carpet napalm, no ground defence is going to be of any use. Furthermore, I'm not an architect so as for the feasibility of constructing these things, I wouldn't know.