Don’t pay attention to the rolls. Just hit the button and then close it up top with the X. The quest will be right there. Spending time looking at the dice roll is just a waist when you could of already been in an AI game.
Get a group of people together with AI boost.
For XP Contribution games - Use TLV to just soak lanes. can easily get this done in 1 game on adept or 2 games on beginner.
For Siege use Hammer ( or any other specialist. knock down everything including the walls/wells/structures. Harass heroes off and go for clearing the wave as you get the contribution for that).
Multi kill is easy. Just mob with the rest of the team on AI beginner
The only one that is the longest for sure is the Win 3 games. For the other ones that want you to win with a support or a warrior, just use your lowest level support or warrior so you can get the extra boxes for leveling them
In fact set it up so your list displays heroes close to leveling and just use them in order to complete objectives. If you are partied with 4 other people who have boosts you can generate about 10 boxes an hour this way.