Although movement in 5e is not governed by squares, they are probably a helpful model to this particular exercise. Everything in the universe is effectively measured in 5' increments and using squares to model this will give us an effective answer to this question Let's posit a 50' wall
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Now let's look at the veritcal space next to the wall
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Finally we need to have the flat part on top of the wall, and the space above that
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yxxxxxxxxx
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Now, supposing our hero starts at the first y (marked a below), and runs up, he runs out of movement at the 10th y (marked b below).
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bxxxxxxxxx
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ax
If you start at a, you might be able to climb up onto the top of the wall. However, if you have a 45 foot wall you are safe. If you have to move to a, a dash would be required to not fall (as you don't have the ability to stay vertical).
The problem here is that if the wall extends at all into the square above the one marked b, then you're in trouble, you have to move into the square above it as diagonal movement is not possible.
This will be both at the discretion of the map makers (if the building extends a bit above the square you're probably not going to move diagonally) and also your DM (he may allow a bit of wiggle room here). Consult your DM before you attempt any 50' wall climbs.
45' wall climbs (and lower) are safe though.
There are two different things happening here: your movement speed being halved, and spending half your movement speed. These work differently, and the order matters. The end result is that the grappler can't move-drag after standing up from prone. Here's how it works:
You start with your full movement speed.
Let's use 60′ for the sake of example.
Standing up costs half your movement.
Your current speed is 60′, half of which is 30′. To stand up you spend 30′ of movement. Your speed is still 60′.
Dragging while grappling halves your movement speed. Your movement speed this round has now been reduced to exactly how much you've already spent, so you will have 0 feet of movement to spend.
Your current speed is 60′, but attempting to move-drag a grappled opponent changes it to 30′. You have already spent 30′ of movement and have zero feet left to spend.
The end result is that after standing up, you still have half your movement left, but as soon as you try to drag a grappled opponent you will have no movement left and remain where you are, so realistically you won't bother trying to drag that round. (You could still end the grapple and move your remaining half movement though, of course.)
Best Answer
35 divided by 2 equals 17.5
Nowhere in the rules does it state that movement has to be an integer multiple of 5 feet. There is nothing wrong with moving 17.5 feet, i.e. 17' 6".
But 7 divided by 2, rounded down, equals 3
The problem only comes in when you play on a grid. The variant rules for playing on a grid are found on page 192 of the PHB, and they state:
Hence, 35 feet correspond to 7 squares. 7 divided by 2 equals 3.5, but half squares are not accounted for in these variant rules. Therefore, you will have to round. By default, you always round down, unless a specific rule says differently. Hence, standing up from prone costs 3 squares of movement, leaving you with 4 squares of movement for your turn.