The numbers that fly out are the hitpoints decreased in the enemy. Green numbers mean hitpoints healed. Hitpoints are displayed in the unit information, but if a unit dies, the hitpoints for the next unit is shown.
The yellow with red means that you scored a critical hit, double damage! Happens occasionly depending on abilities and Morale.
Go to options in the main menu and check if you have autocombat enabled. Otherwise, the enemy troops could have more initative than your troops. The enemy then have their turns before your first.
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