As to exploring: Yes. You can explore you locations as many times are you are able to do so with the cards you have. There is no limit. You get 1 free exploration and then can play cards/activate effects/use powers to continue exploring after that. As long as you have another card/effect/power, you can explore.
As to taking damage, I'm not quite sure what you are asking, but here is how taking damage works. Let's say Harsk failed a check to defeat a monster by 3 and has to take 3 damage. He reveals a Magic Shield to reduce that by 2. That leaves him 1 damage to take, so he discards a card from his hand (it could be the Magic Shield if he chose to, because by revealing it, it is still in his hand.)
FYI: Here is a link to a guide I made. In it I try to give lots of examples of things, like taking damage and exploring and playing cards. It might help with some of your questions, it might not. But check it out if you wish. It obviously isn't an authority, but I'm hoping it's at least helpful.
http://boardgamegeek.com/filepage/101370/a-guide-for-new-players
Hawk pretty much explained this, but I will talk about the second question. Hawk is describing combat damage, but there are also cards that will deal damage before and/or after the encounter. You may follow the same basic rules, because this is really a small clarification.
When you take the combat damage, you may play armors to reduce the amount of cards that you will need to discard. You do not need to discard the cards first and then play the armor to get those cards back. You will never have to discard them in the first place.
The damage that happens before or after an encounter happens after evade checks.
So if you Evade an encounter then you also Evade the 'do this before the encounter' conditions.
If you are supposed to take damage from combat you can use items like your shield that lower combat damage.
Question for Hawk/Erixian
If you are dealt a specific type of damage Acid or Fire do you need a specific defense item/armor to block it or can you use the elven chainmail to block it if you use its primary to block some or the secondary option to block all damage by recharging it?
When dealt damage, you can use any card that says "reduce all damage" (or simply "reduce damage") and doesn't say what kind of trait or any card that reduces damage of the trait of what you were dealt.
So the Elven chainmail's banish/bury power can be used to reduce Acid or Fire damage since it says "reduce all damage". But you couldn't use its first power because that only reduces combat damage.
When dealt damage, you can use any card that says "reduce all damage" (or simply "reduce damage") and doesn't say what kind of trait or any card that reduces damage of the trait of what you were dealt.
So the Elven chainmail's banish/bury power can be used to reduce Acid or Fire damage since it says "reduce all damage". But you couldn't use its first power because that only reduces combat damage.
All of this.
Most armor will tell you that it can be used to prevent combat damage, whole some higher level armor will also allow you to prevent specific types of damage. There are also items which allow you to prevent specific types of damage.
Exploring - Don't forget to read your location's power. Some locations allow you to explore again if you encounter a certain kind of card on your first (free) explore.
So, let's see if I can come up with an example from memory (my cards are not available to reference.) Let's say you're Harsk at the General Store.
Explore 1 - Free explore. You get this as part of your turn. You encounter an ally.
Explore 2 - Free, given by the location card because you encountered something other than an item, armor or weapon.
Explore 3 - You discard a Blessing to get this explore.
Explore 4 - You discard an ally that has a power of "discard ___ to explore again."
Etc. and so on, as long as you have cards that give you the power to explore.
Note, other players/characters cannot give you another explore by discarding blessings or using other cards, unless the power specifically says it allows another character to explore (and I don't know of a card that does that.)
Note also, there are also ways to examine decks. A Spyglass will allow you to look at the top two cards and put them back on top of the deck in whatever order you like. Also, Harsk has a power to allow him to look at the top card of his location at the end of his turn. Seelah can look at the card before her turn, with the conditions that she has to put boons on the bottom of the deck.
I thought you were only allowed one additional exploration each turn?
Page 9 in the rulebook reads "Many effects allow you to explore again on your turn; if, during a single exploration, multiple effects each give you an additional exploration, it counts as a total of 1 more exploration, not a series of additional explorations."
This is for when multiple things trigger at the same time. For example, if Ezren acquires a magical item for his first explore at the academy, his own ability and the academy's ability would both grant him an explore. This rule simply states that he can't "bank" an explore by exploring for one of these abilities and then exploring for the other one afterwards.
You can indeed explore multiple times during the same turn if you have abilities that let you do so. Just not if they all trigger simultaneously.