Does my purchased Class Card apply to all of my Imperial Deployment Groups for each Activation Phase or does it just apply to one Deployment Group of my choosing during an Activation Phase?

Specifically, does the following Class Card apply to each Deployment Group during an Activation Phase or just one Deployment Group: " Adaptive Weapons - While an Imperial figure is attacking, you may replace 1 die in its attack pool with another attack die of your choice."

If the card doesn't specify or require it to exhaust then yes.

Does my purchased Class Card apply to all of my Imperial Deployment Groups for each Activation Phase or does it just apply to one Deployment Group of my choosing during an Activation Phase?

Specifically, does the following Class Card apply to each Deployment Group during an Activation Phase or just one Deployment Group: " Adaptive Weapons - While an Imperial figure is attacking, you may replace 1 die in its attack pool with another attack die of your choice."

It depends on what the card says. The one you mentioned:

This is an extremely powerful card because it is NOT assigned to any one group and applies to all Imperial figures through each phase, each round, and each mission throughout the rest of the campaign. Most cards are assigned to a particular group or figure and may be exhausted, depleted, or discarded or used only once per round. This one is not. This indeed is a truly powerful card.

If the card doesn't specify or require it to exhaust then yes.

If a card is Exhausted does it mean it can only be used once per round? Or does it mean it can only be used once per Activation Group?

If a card is exhausted it cannot be used again until it readies which means normally once per round unless some other effect let's it ready outside of the normal time