Learn German: Doch, der Würfel war wichtig für meine Runde! - Yes, it was! The die was important for my round!

Doch is used to contradict a negative statement or answer a negative question with a 'yes'. This sentence is an argumentative reply to something like 'The die wasn't important, was it?'. If you used Ja here, you would actually be agreeing with the negative statement and saying the die was not important!

Is Würfel singular or plural here? I thought the word was the same for both.

You are right that the noun itself looks the same in both singular and plural (der Würfel / die Würfel). We can tell it is singular here because it uses the singular masculine article der and the singular verb war (was). If it were plural (the dice), it would be die Würfel waren.

Why is it meine Runde with an '-e' at the end?

The preposition für (for) always forces the following noun into the accusative case. Because Runde is a feminine noun (die Runde), the possessive word 'my' needs the feminine accusative ending, making it meine.

What does Runde mean in this context?

While die Runde can mean a physical circle, a lap, or a round of drinks, in the context of board games (which fits perfectly with der Würfel, meaning 'the die'), it means a 'round' or 'turn' of play.