Connection fails just after winning a battle in Adventure Mode. How can I know if I received the cards?

Check your card collection for the Mad Scientist card. That is the reward from the fight if you completed it. If the next fight is available in adventure mode then that also indicates that you completed the previous fight. You just missed the animation/notification of the cards you get/got.

You can tell by the big green V on the boss' icon. If it's marked as completed, you will always get the cards.

Your card collection is stored on Blizzard's servers, not on your local device. Your client freezing will not prevent you from getting the rewards. As long as you can see the match ending, it's safe to assume that everything is in place.