Two quick questions

1. How do the times for speed runs work? Is it simply the cup completion time that you’ll see in records or does it include time in between races?

2. Can I submit a personal best with a picture of my time as validation, or is a stream required due to the above difference in time recording?

After reading through a few other questions, I get the difference between speed running and a normal time trial. If my time from all cup is a 34:04 in-game, could I still submit an estimate for speed run?

1. The timing is displayed in the rules per category, but let me explain it here as well: You start the RTA timing when you start the cup. This means you first see the red box "OK?" at the top, and then you press A again and start the cup, thus also starting the timer. Timing ends on when you cross the finish line in the last track of the cup.

In Game Time (IGT) is the final time displayed by the game at the end of the cup. The reason we also use RTA is to make sure everyone is playing 60hz and is not taking any breaks between gameplay.

2. Yes, as the rules display only top 10 runs require video. I've been thinking about changing this but for now this is still the case.

As for your other question, it depends. If you know for a fact that you mashed A all throughout the all cup tour, feel free to add ~6m and 20s to your IGT to get your RTA. If you're not 100% sure, I'd recommend just doing another run and making sure the timing is correct.