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:

1. You start with your full movement speed.

   Let's use 60′ for the sake of example.

2. 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′.

3. 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.)

A creature usually can't take a 5-foot step to enter a square that hampers the creature's movement

When determining the movement costs for hampered movement, the game usually doesn't care where the creature is and instead cares where the creature's going:

Difficult terrain, obstacles, and poor visibility can hamper movement (see Table: Hampered Movement for details). When movement is hampered, each square moved into usually counts as two squares, effectively reducing the distance that a character can cover in a move.

If more than one hampering condition applies, multiply all additional costs that apply. This is a specific exception to the normal rule for doubling.

In some situations, your movement may be so hampered that you don't have sufficient speed even to move 5 feet (1 square). In such a case, you may use a full-round action to move 5 feet (1 square) in any direction, even diagonally. Even though this looks like a 5-foot step, it's not, and thus it provokes attacks of opportunity normally. (You can't take advantage of this rule to move through impassable terrain or to move when all movement is prohibited to you.)

You can't run or charge through any square that would hamper your movement.

Emphasis—and extra emphasis—mine. Thus at issue is the creature's destination square not its current square. The creature's already in its current square, so the creature doesn't have to care about that square's terrain, having already overcome the difficulty of entering it!

For example, going from a square of supernatural darkness into an adjacent square of bright light is just normal movement, but going from a square of bright light into an adjacent square of supernatural darkness usually costs double movement due to poor visibility. In fact, going from any square into an adjacent square of supernatural darkness usually costs double movement due to poor visibility, the condition of the starting square typically not mattering at all.

Exiting then reentering the same square of hampering terrain still costs additional movement, of course. A creature can't move normally back and forth between squares of hampering terrain just because it's occupied the same square of hampering terrain previously. For example, moving from a square of supernatural darkness to a square of supernatural darkness then back to the original square of supernatural darkness does not speed up the return trip.