1. Now I know that you can activate quick plays from your hand, but what exactly does that mean? Could I play a quickplay from my hand if all my M/t zones are full? Or could i do the following.

Player A attacks Mobius the Frost Monarch with Chaos Sorcerer.
Player B has no response.
Player A activates Rush Recklessly during the damage step from their hand. Is this possible.

2. Secondly, I know about you not being able to chain to Counter Trap cards with any card except other Counter Trap cards, however how far does this go? For example:

Player A attacks Player B directly with Don Zaloog.
Player B responds with Sakuretsu Armor
Player A responds with Solemn Judgment,

Does the attack go uninterrupted or is Player B allowed to chain a second Trap Card, like another Sakurtesu Armor?

he is not able to respond to the attack again, but he may still activate trap and quickplay magic cards while still in the battle step. The chance to respond to the attack is lost however. This means no sakaretsu, mirror force, or cards like them. You COULD still play waboku, book of moon, or gravity bind.

1. To activate ANY card, you must have an open zone to be able to use it unless it activates somewhere else.

In this case, you DO need an open m/t zone to be able to use a Magic or Trap card at all. So in your first scenario...if you don't have any open Magic or Trap Zones, you could not use Rush Recklessly.

Otherwise, yes, you can activate it from your hand during the Damage Step since it's your own turn and it's a Quickplay Magic that directly modifies the ATK/DEF of a monster.

2. Read agent's post.