What determines the horse’s sprint duration and speed

If you are within about 1-4 blows of killing an opponent, a "finishing move" animation may appear, depicting one or several fatal strikes from either a first-person view or third-person "KillCam". If the Savage Strike (One-handed) or Devastating Blow (Two-handed) perks are unlocked, there's a chance that this will be a decapitation. Note that under the right circumstances, NPCs and Dragons will also be able to perform finishing moves, both against each other and against you.

^{Source: UESP wiki, "Skyrim: Combat" article}
Finishing / Kill moves are not completely random. Finishing moves are affected by the in-game variables, 'KillMoveRandom' (default: 50%), and 'DecapitationChance' (default: 40%) and various factors, some of which are:

• Kill Cameras will only be performed if killing the enemy will take the player out of the combat mode. Due to this, killing the last active/aggressive enemy will often trigger a finishing move / kill cam. But this does not mean that only killing the last active/aggressive enemy will trigger a kill cam, e.g. killing an enemy while sneaking and remaining undetected will allow a Kill Camera, even if there are more enemies present.
• The opponent being killed. Some enemies in the game seem to have no finishing moves associated with them.
• Your position relative to the enemy. For example, there is a one-handed sword stab-through-the-middle execution that will only play if you're standing behind a humanoid opponent. There are also finishing moves which require your enemy to have surrendered and fallen down to their hands and knees.
• It should be noted that your steath state has an affect as well. E.g., you can cut throats with a dagger.
• The terrain: finishing moves appear to depend on relative elevation. While it's possible to enter into a finishing move animation on uneven terrain, it's far more likely to happen if both you and the enemy are standing on a flat, level surface.
• Other nearby obstacles: If the enemy NPC is too close to an object like a table, it may interrupt the animation and leaving you standing there until enough time has passed and the game gives you back control of your character.
• Chance: Some animations seem to play more often than others.
• Slow Time: Finishing moves will not trigger while under the effects of the dragon shout Slow Time.

^{Source: UESP wiki, "Skyrim: Combat" talk page (edited to take into account the comments in reply to the original post) | TES wiki, "Kill Camera" article (edited to explain further)}
You can modify the finishing move-related variables using the console commands, set killmoverandom to <0-100> and set decapitationchance to <0-100>. Or, use mods like SkyTweak (configuration page screenshot), The Dance of Death - A Killmove Mod (configuration page screenshot) or Killmove Control which have UIs to help configure those settings. 'The Dance of Death' mod can even remove the 'last active enemy' restriction.

This is an issue many have experienced.

In the console, type the following:

set playerfollowercount to 0

Now the game will recognize that you no longer have any followers.

note: this only works on PC.