Speed is a game noun
The basic rules define speed as:
Every character and monster has a speed, which is the distance in feet that the character or monster can walk in 1 round.
On your character sheet this is probably listed as "speed: 30ft" or something similar. Anywhere the game says "speed", it is talking about this speed. You use it for standing up, swimming, climbing, etc.
An affected target's speed is halved
When standing up from prone:
Standing up takes more effort; doing so costs an amount of movement equal to half your speed.
Now that your speed is halved, when you attempt to stand up from prone it will cost half of your new speed. If your original speed was 30ft, then it was halved to 15ft, standing up from prone will cost 7.5ft.
What if I am also hit with a Ray of Frost spell while slowed, reducing my speed to 5 feet for 1 round? Am I able to stand up at all?
If your speed is 5ft, then standing up costs 2.5ft. That leaves you with 2.5ft left over. If you are playing with the grid variant rule, then the DM will probably rule that you should round down your remaining movement and you cannot move 1 square.