1. If you are engaged with multiple attackers, can you heal after each attack or do you have to complete all engaged attacks before your action?
Assuming you have actions that let you heal, you may use them between attacks. (After Step 4 in the Combat description, and before Step 1 starts for the new attacker.)
a. If I understand the steps you can only use an action after dealing shadow cards but before flipping the shadow card?
No, you can use an action then, but you can also use actions between resolving the shadow card and dealing damage.
b. Then you can use your next action after all Creature and Player attacks are complete before moving to Refresh Phase?
2. If you do not attempt the Quest phase do you still draw a card for the staging area, or just skip that step all together?
You cannot skip the quest phase. If you don't commit any characters, you still reveal one encounter card per player, then raise each players threat by the total threat shown in the staging area (since your total Willpower will be a big fat zero). This is almost always a BAD idea.