Hello! The RCDs are a fantastic regiment steeped in much history, but they did not manouver mounted until 1916, and truly only had 'from the saddle' opportunities in 1918. In their particular instance, they would not have used heavy mounts, as the tradition of the Canadian militia would've made dragoons much more akin to their original role of 'mounted rifles' rather than catch-all cavalry, as in the Napoleonic wars. Generally speaking however, no true 'heavy' cavalry existed in the British tradition of this time, but rather 'cavalry-of-the-line', that is, Dragoons and Lancers, which were expected to be of the most flexible variety. The idea of 'light' and 'heavy' Dragoons would be antiquated by the Great War, a better phrasing of the question is 'what is the difference between mounted rifles, and Dragoons.' The difference being (and this is far from absolute), that one is trained to fight from saddle with pistol and sabre, and the other is only truly able to fight on foot. Experiences in the Middle-East really showed that the two were often interchangeable, with many 'mounted rifle' units fighting from the saddle with skill and courage. Two excellent books I reccomend to you on that campaign are: The Last Crusade: The Palestine Campaign in the First World War by Anthony Bruce, and How Jerusalem Was Won by W.T Massey. The latter is a near contemporary publication, and if you can handle/dismiss the obvious positive tone towards the Entente forces, it is the history to read on the campaigns, which have arguably the most 'mounted' and cavalry combat of WWI.

In terms of what 'British' (and I'm going to stretch this term to the Empire at large) units were mounted on, they were generally of lighter stock, not the most ill-tempered beasts, but still Stallions of impeccable form. A lot of this has to do with British Cavalry tradition and how its role compared to the predominant infantry in the military ethos. A type of Horse I can name to you off the top of my head that saw heavy use is the Australian 'waler' breed that mounted many of Australia's light horse divisions in the Middle East.

However, we know that horse breeds used for units meant to fight from the saddle were still often of greatest stock, and the British deliberately started a campaign of thorough-breeding in order to mount regiments on tall and imposing Stallions. The book The War Effort of New Zealand by a gentleman named Reakes, goes into further detail. In it he asserts that the NZ government found that overly-tall horses (above '15 hands') were strong and imposing, but were a strain on supply (requiring quite a bit of fodder to keep in firm fighting condition). 15 Hands therefore became the base line size, the ideal. This struck a balance between endurance over long distances as well as strength and height.

Needless to say, this was peculiar to the mounted arm. We can expect units that fought dismounted, but marched mounted, to have gone below the 15 hands threshold, focusing more on endurance than strength, being a means of transportation. On the other end of the spectrum you would have something like Draught horses, hardy working animals more akin to farm work, frequent feed, and good care, to help with the manhandling of artillery and machine-guns. Draught Horses would not be as imposing a supply strain because they generally supported units that were not high in manpower, and therefore could spare more supply space for ammunition and fodder. Dragoons would, if you allow me an educated guess, across almost all nations, attempt to strike the ideal. The very nature of a Dragoon unit was to fight with equal competency mounted, an dismounted. This means a steed had to be able to march long distances, and still be able to carry home a charge - albeit with very much reduced ferocity, as opposed to a French Destrier - with a sufficient level of shock.

I've touched upon it briefly in the previous paragraph, but to very quickly answer your second question: No, at least until the 'era of maneuver' (1914-1915, some will say 1916) ended and the era of trench warfare began. These ideas of cavalry were not uniform across Europe or indeed the world. While all nations had cavalry of the line, or had units that blurred the distinction between light and 'of the line' (e.g: Hussars), many entered the war with cavalry formations that may have seemed more in-tune with the wars of the 1850s-70s. France still retained Curassiers for shock action, the Germans and Austrians still had Ulans (Lancers influenced by Polish tradition), and the American cavalry (I know, I know, not Europe) was very much based in the findings of the Civil War, and were probably the closest to what Cavalry would appear like by 1918. Units like Curassiers, utter anachronisms by 1914, rarely survived past 1915, and were either reimagined and rearmed, or hurled into battle as infantry. As an addendum, most of the US Cavalry units that shipped out with the expeditionary force in 1917 either fought as infantry, or curiously, machinegun battalions.

Edit: Book recommendations! The second question.