Movement from a special ability is still reduced by prone, etc.
To address the last point first, I think it'll be helpful to compare the basic rules about movement to this special ability.
According to the first part of the Movement and Position rules (PHB p.190), which is the basic rules of moving:
On your turn, you can move a distance up to your speed. You can use as much or as little of your speed as you like on your turn.
And Instinctive Pounce says,
As part of the bonus action to enter your rage, you can move up to half your speed.
Both rules use virtually identical wording. I don't see any possibility that they were intended to be spent differently. Movement, it seems, is just movement, even if it comes from a special ability. You spend it the same way in every case, and all the usual cost increases will apply -- such as from difficult terrain or being prone -- unless the source of movement specifically says you can ignore the cost increase, or if you're using a movement mode that lets you ignore it (such as flying over some kinds of difficult terrain).
The rules for difficult terrain and being prone, on that same page, don't care what your speed is, only that you're using movement:
Every foot of movement in difficult terrain costs 1 extra foot.
Every foot of movement while crawling costs 1 extra foot.
Whether it comes from your speed or an ability, it's all movement, and it functions as such, subject to all the usual rules.
So if you're prone, you can use your 20 feet of movement to crawl 10 feet, provided you don't have another effect (like difficult terrain) also increasing your movement costs.
Can you stand up? Yes.
Again, movement is just movement; there isn't a special kind of movement that can be used to stand up and some other kind that can't. If you can use "you can move up to your speed" to stand up, then you can use "you can move up to half your speed" to stand up.
The rules for Being Prone say:
Standing up takes more effort; doing so costs an amount of movement equal to half your speed. [...] You can't stand up if you don't have enough movement left or if your speed is 0.
Now this refers to your speed, not to how much movement a special ability granted you. It will always cost half your speed (20 feet, in your case) to stand up. If you get to move half your speed from Pounce, then you can use that to stand up, but that takes up the entirety of what Pounce gave you.