So, I know that Dice are a "limited resource." But let's say that I have 8 dice in my pool and my opponent has 3 dice. And in the second round of combat, I play a card that lets me add a die to my pool. Am I allowed to go over the 8 dice limit in this case? Thank you in advance for an answer to this question.

No. The 8 dice limit is absolute and cannot be overridden by any game effects.

Thank you for the clarification.

Of course, it is possible to spend a die for something and then later gain another die to put you back up to eight. You'll just need to follow the proper order to pull that off.

You'll notice that there are only 16 dice in the box

If you already have 8 dice and you play a card that let's you gain a dice are you allowed to discard one of your other ones in order to roll a new one?

No

If you already have 8 dice and you play a card that let's you gain a dice are you allowed to discard one of your other ones in order to roll a new one?

If you were allowed to do so, it'd be clearly stated in the rules

a follow up question about gaining dice.

Let us say that on the first Execution round I play a card that lets me add 1 more dice. Let say I have a total of 3 dice before playing that card that let's me add 1 or 2 more dice.

My question is will that 1 or 2 dice added on the first execution round by playing a combat card carry over to the 2nd and 3rd execution round?

That is perfectly fine, as dice are not removed like combat tokens without an ability saying so. In general the permanence of combat boosts is tokens -> dice -> icons, from least to greatest.