This answer, written a long time after this question was last addressed, contradicts the Accepted Answer which I submit is in error, as you will see.
Executive Summary:
The amount of distance a grappler can move after standing from prone is half their available movement, because standing consumed half of their initial available movement. Grappling does not change the rules related to standing from Prone.
Their remaining available movement will be modified by the grappling rules if they do not release the victim from the grapple.
Scenario 1
“The grappler has a speed of 60 feet. Standing up takes 30 feet. They
stand, and only have 30 feet of movement left. Having someone
grappled, they can only move 30 feet in any direction with the one
they've grappled -- but they also have exactly 30 feet left. So they
can move 30 feet in any direction with the one they grappled, so that
being prone and grappling someone has no effect on you.”
Answer
No, intuitively not, because this would have them moving 30 feet forcing a grappled creature along with them, without suffering any movement penalty.
Let's look under the hood....
Grappler has speed of 60. He stands, using up half of his available movement. 30 feet remain.
Moving while grappling is covered by the Rule “speed is halved” (Chapter 9, page 77).
What do the Rules mean by “speed”? Which speed is halved, the 60 speed written on your Character Sheet or the 30 currently available to you?
Your speed as defined by the Rules is the distance in feet that the character or monster can walk in 1 round (Chapter 8, page 66). Or to paraphrase, the amount of movement available on your turn.
But we also know that the amount of movement available to you per turn (speed) changes during distinct phases because
However you’re moving, you deduct the distance of each part of your
move from your speed until it is used up or until you are done moving.
(Rules, Chapter 9, page 73, Movement and Position)
They realised that our speed might be modified differently, at different parts of our turn's movement.
After you have completed a "part of your movement" any subsequent part is distinct and separate and subject to whatever new conditions might be upon you. If part of your movement takes you through Difficult Terrain, only that part of your movement is penalised. If part of your movement takes you through the Area of Effect of Spirit Guardian, only that part of your movement is affected. If part of your movement has you penalised because you are standing from prone, only that part of your movement is penalised. And if part of your movement is used 'grapple-moving' only that part of your movement is penalised.
Putting it in another way, of your 60 feet of movement, you used half of it standing from prone. Were you moving while grappling as you did that? No, of course not. So the speed that is halved "when you are moving whilst grappling" excludes the first part of your movement which was not subject to that condition.
So, after you stand up from Prone, you deduct the distance of 30 feet from your movement, leaving you with available movement (60 minus 30) of 30. That part of your movement is completed and the appropriate amount has been deducted from your available movement.
Now the second part of your movement commences. Available movement (speed) is now 30; we halve it for the answer 15 and this is the distance in feet we can move a grappled creature in this scenario.
Scenario 2
“The grappler has a speed of 60 feet. Standing up takes 30 feet. They
now have 30 feet left. Since they have someone else in a grapple,
their speed is halved to 15 feet, and they can move with the grappled
creature 15 feet.”
Answer
Yes, as above.
Scenario 3
“The grappler has a speed of 60 feet. Since they have someone in a
grapple, they only have 30 feet of movement. Standing up requires 30
feet of movement. So when they stand up, they have no movement left.”
Answer
Absolutely not. They have a speed of 60, and it only costs them 30 feet to stand up. They have 30 feet remaining and it didn’t just evaporate simply because they are holding someone’s hand.
Unfortunately, this is the position put forth by the Accepted Answer. In addition to my analysis above, pointing to a different answer, I have shown at the end what I think is perhaps the flaw in that author's reasoning.
To finish off with your final question:
Scenario 4
“Also consider the case when the grappler has the Athletic feat, allowing them to stand up with only 5 feet of movement instead of half
their total.”
Answer
This plays out as follows: Your speed is 60. Standing costs 5, leaving you with 55 feet remaining. Should you choose to move without releasing your grappled victim, you can move with them a further 27.5 feet. Rounded to 25 I suppose, if using the grid rules variant at Chapter 9, page 74.
Why the Accepted Answer is Incorrect
(with apologies and all due respect to @SevenSidedDie)
It assumes that you have 2 speeds, a normal speed of 60 and a "grappling speed" of 30. This is a fallacy. Half speed or "grappling speed" are not speeds, in the same way that walk speed 30 and flying speed 60 are speeds as indicated by the Rules at page 73 "Using Different Speeds". You are not "switching to grappling speed" in the same way that you might switch from walk speed to fly speed. So you cannot say "now that I am grappling my new speed is 30 and since I have moved 30 already I can no longer move".