Hello,
it’s a Common Cathode version if that matters.
Yes that does matter as it changes how you control the digits.
// pins 10 and 12 turn on the whole digit: HIGH is on
From this I assume pins 10 and 11 are connected directly to the common cathode of each digit?
If that is the case, as the cathode connects to ground, LOW would be on not HIGH.
And the same is true for the Anodes, they connect to positive supply, so HIGH is on and LOW is OFF.
Although having said that, the code seem's to be in the correct order, just your comments are wrong.
// Turn the Digits off
digitalWrite(10, HIGH);
digitalWrite(12, HIGH);
// first digit, turn top bar on
digitalWrite(7, LOW);
digitalWrite(2, HIGH);
// Turn on the Digit
digitalWrite(10, LOW);
// sleep
delayMicroseconds(10000);
I'm not sure about the leakage, but if the digital output for the segment that is off, is still turning off when the digit is turned on, it may appear feint.
I would try adding a very small delay between setting the segments and switching the digit on.
Or of course wait for a more conclusive answer from an expert 
Just what I would try while waiting 
Dean