Sires whose progeny improve from 2 - 3

I would put most of the staying sires on in that category although Sir Percy seems to get 2yos. If you look at my table of sires based on number of races won, it is dominated by sprinter/milers who are able to get winners at all age groups. It is hard for staying sires to make the top ten because they get few 2yo wins. The dam is important as well of course and so is the breeder. The Aga Khan gets few 2yo wins but his strike rate with maiden 3yo's is close to 50%.

Look at the Equinome website.  That will give you an idea of the likely distance ability of sires, and then of their offspring.
Two year olds seldom run further than a mile.  So 2yo sprinters, middle-distance, and stayers are mostly limited to 5f,6f,7f,8f and it is difficult to separate them on distance suitability.
When they run at 10f and further as 3yos you hear comments like "it didn't train on".  What actually happened is they were tried at a distance beyond their pedigree.
Good example of this were Dawn Approach in the Epsom Derby, up from 8f to 12f, and Toronado stepped up from 7f/8f to 10.4f in the Juddmonte International (they immediately brought him back to 8f for his next (and last) four races).
If you like you can e-mail me at "my user name" at eircom dot net.

I think I can point you to races where people may make mistakes.  These races are small in number, and the mistakes may be small in number.

I assume that by a "mistake" you mean running a horse over the wrong trip. Sir Mark Prescott seems to do it as a matter of policy with horses destined for handicaps as do many other less obvious trainers. When it comes to trying a top class horse over a different trip, it is moe often an experiment than a mistake. How would you describe Tiggy Wiggy at Newbury? There are few enough opportunities for a 3yo sprinter so every year, sprinting 2yo's run in classic trials in case they stay. Henry Candy has clearly decided that Limato won't stay but he is now limited to all aged sprints in order to progress.

Limato is a gelding that rules out the classics anyway.
I am sure if he was a colt they would have at least tried to see if he stayed as you correctly pointed out Sageform.
It would be interesting to know who made the decision to geld him and why.
I am assuming the fall out is why Henry Candy ended up training the horse in the end.

I had overlooked that. It might help him become a champion sprinter for a couple of seasons as there will be no fear of getting him beaten and prize money is the priority. It is highly unlikely that a 5 furlong specialist would stay a mile anyway.