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I ask because I love the idea of using Elysian rules to make a Cadian Drop Troops regiment. I know that Cadians train in all aspects of combat so they know how to deploy from Valkyries or other drop ships, but are there any regiments dedicated (other than Kasrkins of course) to dropping Cadians from flying things?

Cadians can be formed into nearly any regiment. So yes, there will probably be some airborne Cadian regiments out there, same as there are airborne Catachans or Vostroyans or anything else.
Cadia pumps out so many guardsmen, there's bound to be an aerial unit out there.

I think the Shield of Baal book had a Cadian airborne regiment that used Valkyries.

Edit: Found them on the lexicanum (http://wh40k.lexicanum.com/wiki/Notable_Cadian_Trooper_Regiments) 1433rd Airborne

Honestly, outside of a few very specific worlds almost all planet specific forces field most(if not all) regiment types.

Catachans are not likely to field a mechanized infantry force, and Kreig troops are even less likely to have jungle fighters; but these specialist "world of hats"-types are more the exception than the rule.

Harakoni actually have an armoured and an artillery regiments for example, which would be odd if they were pure drop-infantry like the ellysians seem to be(even they are likely not drop-only as well)

The thing with the Elysians is that their non-drop-regiments are essentially indistinguishable from Cadians, so nobody bothers with them.

I'm sure I remember seeing a vehicle from a Catachan armoured regiment in one of the Imperial Armour books; possibly IA1 1st edition. It was noted as being rare, but it's still there. Krieg produces armoured and super-heavy tank regiments, and rough rider regiments as well as their more famous siege infantry regiments.

There's even at least one Imperial Guard "regiment" which consists of Thunderbolt fighter squadrons - the Phantine XX. Thanks, Dan Abnett.