Vikings didn't use horses for two very separate reasons:
1.) They did not come from a culture of equestrians.
2.) Horses would not have fit very well on a longship. Horses had a hard enough time making sea voyages on much larger vessels. No way a longship was going to carry enough horses even for the elite to utilize.
As for why the culture Vikings came from (Vikings were raiders, not a people unto themselves) didn't utilize horses - they just didn't. Part of it was certainly a societal view of horsemen as 'away from the combat' and therefore unmanly. Part of it was military tactics (a favouring of the shield wall and two-hand axes, neither of which are suitable for mounted maneuvers).
There was also a distinct lack of battle-worthy horses in the European North. Even during the early period of the Normans - horses used in warfare were only passable, barely more than ponies. Most early medieval warhorses had bloodlines tracing from the Middle-east.
As for the Normans -- they are, in fact, largely accepted as the first real knights. That is not to say, however, that there weren't precursors to the tradition of the mounted soldier. There were many, tracing back well before Charles The Great. The Romans, even the Ancient Greeks, had armoured, mounted swordsmen/spearmen.
The Normans get the credit largely for really pushing the tactic of the heavily-armoured mounted warrior. They improved it, made far greater use of it, etc, than anyone to come before them in all of Europe. Some scholars also agree that they were definitely the first to use the couched lance technique that 'knights' would become famous for.
They adopted French customs and methods of warfare, but improved on the mounted aspects that had been mostly stagnating in that region. Following the line of history, it would seem moreso that the French adopted their mounted tactics to become full-fledged 'knights' from the Normans.
The Normans really are a fascinating subject. Such a culture. Magnificently heartless, inventive, ruthless, and powerful. Their military capabilities were really quite astounding, when you study them more thoroughly.
Forge not works of art but swords of death, for therein lies great art.
"The Gods made heavy metal and they saw that it was good." - Manowar