Basically yes to all of your questions.
If I understand the RAW correctly, a character could...
1) get on a horse and ride away on it in the same round
they could also
2) do 1 and ride away (via the horse disengaging) from adjacent enemies
they could also
3) stand up from prone and combine with 1 and or 2.
4) get on that horse, attack, and ride away. And combine with 2 and or 3.
Is that all correct? Does combining all just get too cheesy?
This is covered under Mounted Combat (page 76 basic rules)
When it comes to mounts and combats, there are two basic options. Do you control the mount or they can act independently. Intelligent mounts (i.e. most dragons, paladin's horse) act independently.
If the PC is controlling the mount, they both go on the PC's initiative and the only thing it can do is move and take the actions Dash, Disengage and Dodge.
If the mount provokes an opportunity attack, the attacker can target the PC. If the mount dodges, there is no OA.
Mounting requires half your movement, your movement limit does not count against the mount's movement. For #4, you'd have to be prone next to the mount when you stand because you don't have any extra movement to get to it.
If a mount is acting independently it gets it's own initiative*, attack and movement and is basically an NPC largely under the control of the player. The player is just along for the ride, and the DM gets to decide the mount is going to follow orders or not. Not normally an issue with special bonded mount like paladins get, but certainly can be an issue with other creatures.
So some of the things you mention may not be possible with an intelligent mount based on initiative and sequence.
Also remember that the DM is free to control the mount - a warhorse may not sit there patiently waiting for you to stand up, it may decide to flee combat for example.
*For simplicity, I do have the mount and rider go on the same initiative. It does make the paladin's mount slightly more effective, but not so much that it's overpowering.