Getting back up after getting knocked down: How to recover quickly and get rid of the dizziness?

If you are sparring hard and you get hit, you don't have the time to pause and call your family doctor. I want to know how experts (i.e., those who have serious sparring experience) deal with this.

I'm an advocate for training hard, sometimes really hard. But in doing so you shouldn't lose perspective - there is no point continuing the hard training if it is going to result in semi-permanent damage or an injury that is going to take a long time to recover from. There is no shame in taking a knee (or staying down) in training, even the best athletes can get hurt. You and your coach are both the best judges of when to pause that hard training, and the coach should be watching and listening to you carefully. Even if you have a death wish your coach has a duty of care they should be observing.
If you are training for the ring then this hard full contact training should be gradual, and your sparring partner should be aware if they've hurt you. If this training is not for the ring then you have to question exactly how hard you need to go - it is questionable how much you need to be smashed over in the dojo to prepare you for real life.

Are there techniques that can speed up the recovery?

Fundamentally, no.
You can delay getting hit again, but you cannot speed up the recovery time, and that recovery time will differ for each person and each blow received (this is part of what makes some fighters better or more resilient than others).

Is moving around and throwing jabs better? Will it stop the opponent from coming in hot?

This depends on the experience (and intellect?) of your opponent. If they're good then they'll know that they've hurt you, and their coach/corner person should also know. To delay you can use as much of the count time as practical, you can evade, you can cover, you can spit your mouthguard, you can do a bunch of little things to delay getting hit again. None of these techniques are new, they've all been used a billion times before so judges and opponents know what to watch out for. But if you're good at it it's possible that your opponent won't know - essentially you need to be good at acting. If I knock someone down and they get back up and continue as they were then I'm more likely to think they're okay, but if they get up and then go hyper I'm more inclined to think they're covering for something.