Note: I showed this question to a friend, oddly enough he gave me a full answer. He refuses to make a Reddit account and insisted that I post this for him. He's highly interested in cavalry units of the modern and post modern era and has done much reading on it. If you have follow up questions, I shall show him your questions and respond for him.

Prior to 1942, the Germans and Polish used their Cavalry relatively similarily; as flexible, general purpose and mobile units. There are key differences, however. Strategically and Operationally the Germans had already connected the dots and begun to see their Panzer forces as the successors to the traditional cavalry missions of exploitation and shock-action. When it comes to emphasis on combat, the Polish (more so out of necessity) would've been more willing to put their horsemen into 'combat' situations. That is not to say that the German cavalryman did not engage in combat action, but when he did, it was on foot and no different than his counterparts in the infantry.

German 'mounted' combat is few and far between during its period of heavy use (39-42), without a doubt there were many smaller mounted actions, but the only real ones of worth occured during the Polish campaign and often against their counterparts. In none of these situations did they come off the better. This again can be chalked up to Germany's more modern doctrine, with focus on dismounted action and a healthy respect for the mass-carnage they risk with any sort of bold mounted movement.

The Polish like wise used their cavalry in conjunction with their armored trains to swiftly remount between defensive positions and fought delaying action after delaying action, often on foot with the most basic of towed heavy weapons. In their case however, they did have isolated incidents where they used sabre and lance against German infantry. One particular case had them routing a German battalion that they came upon quite suddenly, only to be scattered in some disarry when German scout-cars came to reverse the situation (the birth of the pure propaganda piece of Polish lancers dashing themselves to pieces against Panzer IIs).

The Germans were keen to learn from examples, and as to be expected doctrine changed even more radically following the conclusion of the Polish campaign. The last Wehrmacht mounted unit would be mechanized and become the 24.PanzerDivision (destroyed, 1942, O. Uranus) and that would be the end of Horses for anything beyond logistics, petty transport and small-unit reconassaince for the German Wehrmacht. The German SS however retained at this period a single Brigade of cavalrymen, which were used - at first - exclusively for disgusting anti-partisan actions in the Prippet marshes. Remaining cavalry units struck a fragile but useful middle between light infantry and mechanized forces, they could go where infantry went and at a more flexible pace, but without the crushing logistical strain of a light (armored) reconassaince unit.

The SS would eventually expand to (I believe) three Horse Divisions, all of which were used as Anti-Partisan units. The SS Brigade of 1941-43 fought several stunning defensive actions during Soviet counter-offensives, but their doctrine for 'on-the-line' combat remained similar to the 1939 doctrine previously discussed - they were mounted infantry. As a last note, their counterparts on the Eastern Front still maintained huge mounted forces, that fought with incredible success given the modern nature of WWII. The Soviet cavalry is often only mentioned as footnotes, but they operated often as self-contained units, providing their own towed anti-tank, anti-air and pack-artillery units. This anamoly may be written off as simple convience on the Soviet Union's part; they did not lack for horseflesh and where not unlike the Confederate States of 1861 in that they had no lack for trained/natural riders.

Sources: 'Riding East: the SS Cavalry Brigade in Poland and Russia, 1939-1942' by Yerger, Mark C. 1996 Schiffer Pub : 'Achtung-Panzer!': The development of armoured forces.... Guderian, Hanz. 1992 London: Arms and Armour

(Addendum: This is what I'd read if you wished for details about how the Germans began to see armor as a technological successor for cavalry)

'Mechanized juggernaut, or military anachronism?: Horses and the German Army....' DiNardo, R.L New York: Greenwood Press, 1991