[RPG] Do Tabaxi regain the use of their Feline Agility trait if they get up from being knocked prone but don’t move to a different space

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.)

Your intuition is correct. The Dash action simply increases your movement, but doesn't actually increase your Speed.

If your speed were 30 ft. and you decided to Dash, you'd still only spend 15 ft. (of the gross 60 ft. of movement) to stand up from Prone.

If you had an item, effect, or class feature that increased your Speed (e.g. the Boots of Speed, which double your walking speed while active, or the Mobile feat), then standing from Prone would cost more, according to the rules as written.