I did some research of my own instead of waiting for another answer. I played 4 games with my Level 5 Mage (leveling her up to level 8 in the process). In none of the games did I manage to use the whole deck, but I wrote down all the cards that I drew throughout all 4 games.
First, to confirm Resorath's answer, none of the Level 2, 4, and 6 cards of the Mage ever showed up in the deck, but the original 5 cards that the Mage starts with all showed up in at least 3 games out of 4. So I can conclude that none of the later Mage cards are included into the basic deck as you level up.
Second, exactly ten other cards showed up in 2 to 4 games each out of 4, while no other cards ever showed up. So, only 15 different cards ever showed up for the Mage, some twice, making up the deck of 30. That means, it's not random, and the cards are set in for each basic deck. Most likely, they are tailored to each class. (For the curious, for Mage they are: Murloc Raider, Bloodfen Raptor, Novice Engineer, River Crocolisk, Raid Leader, Wolfrider, Oasis Snapjaw, Sen'jin Shieldmasta, Nightblade, and Boulderfist Ogre, plus the original 5 Mage cards)
Both of those conclusions could be very good coincidences, but, eyeballing the probabilities of these coincidences, I firmly believe that those conclusions are correct.
So, I did not experiment with all 9 classes, but I found my answer: The basic decks were pre-made by the creators of the game, and they do not change as you level up. I guess the results for the other 8 can be found the same way.
Secrets are triggered in the order they are played.
In your example above, both will trigger.
However, if the attacking minion dies to the Explosive trap, the Misdirection is wasted. (For the record, I learned that the hard way).
Best Answer
The only way to know a secret is, is to reveal it or through process of elimination figure out what it is not. There is only 1 card currently to negate secrets, Flare.
In order to use process of elimination you need to know what secrets are out there. Each class has a fixed cost for their secrets, so that will give you no information regarding what it is. The other thing to take into account is the game format and the regularity with which each secret could be expected.
If you are playing Constructed in "Play" mode and your opponent uses 1-2 legendaries on you, it is safe to expect they have access to the rare and epic traps; but if they are playing primarily the basic cards it is more likely they may not have the rarer secrets.
If you are playing Arena, there are certain secrets you see more often than others. For example, most players rate Explosive Trap fairly highly as a Hunter and it is a common.
That said, there is also some basic logic you should use when testing a trap. If you are attacking a hunter, be intentional about which creature you attack with (knowing that the trap could be a freezing trap), this can sometimes be used to your benefit. If the trap does not trigger when you attack, then it is not Misdirection, Freezing, Explosive, or Snake... leaving only Snipe. So then be intentional about which creature you play next. Also, you would want to attack before you play a creature in almost every scenario because of the high probability of Explosive Trap or Misdirection.
3 classes have secrets; Hunter, Mage, & Paladin.
Hunter:
Snake Trap (epic) - trigger: attacking minion, effect: summon 3 1/1 snakes (beasts).
Explosive Trap (common) - trigger: attacking player, effect: deal 2 damage to all enemy Characters (includes you).
Freezing Trap (common) - trigger: any minion attacking, effect: return minion to hand & it costs +2 to play.
Snipe (common) - trigger: "Play" a minion (not "Summon", so for example it does not trigger from paladin/shaman abilities or imp master), effect: deal 4 damage to that minion.
Misdirection (rare) - trigger: When any character attacks a hero, instead he attacks another random character.
Mage:
Ice Block (epic) - trigger: mage takes fatal damage, effect: mage doesn't die this turn, stays at 1 life.
Spellbender (epic) - trigger: target any minion with a spell, effect: randomly select a new target.
Counterspell (rare) - trigger: cast any spell, effect: that spell is countered.
Vaporize (rare) - trigger: attacking player with minion, effect: destroy that minion.
Ice Barrier (common)- trigger: attacking player, effect: mage gains 8 armor.
Mirror Entity (common)- trigger: "Play" a minion, effect: summon a copy (battlecry abilities do not trigger for mage's copy).
Duplicate (common - Naxxramas) - trigger: friendly minion dies, effect: Place 2 copies of destroyed minion in hand
Paladin:
Eye for an Eye (common) - trigger: paladin takes damage, effect: after paladin takes damage the same amount is done to his opponent.
Noble Sacrifice (common) - trigger: attacking anything, effect: paladin summons a 2/1 and your attack is redirected to that.
Redemption (common) - trigger: any of paladin's minions die, effect: return to life with 1 health.
Repentance (common) - trigger: you play a creature, effect: it's health is reduced to 1.
Avenge (common - Naxxramas) - trigger: friendly minion dies, effect: give a random friendly minion +3/+2