You can cast with any free hand; you don’t need two. So you can cast the spell in one hand and have the dagger in the other. And drawing a weapon is a move action anyway; if you have BAB +1 you can even do it while moving.
You cannot attack with the dagger to discharge shocking grasp however, simply because the rules allow you to discharge it with a touch attack, or with an unarmed strike, and with no other action. There is an ill-regarded and poorly-conceived line about unintentionally discharging the spell by touching anyone or anything. I strongly recommend ignoring it, as it conflicts the rest of the rules, and allows all manner of broken abuses.
As a form of circumstantial evidence, I point to all the variety of options for allowing weapon attacks to discharge spells: they all involve particular investment on your part. They would serve no purpose if you could do it without those investments. For examples, the spellsword class’s Channel Spell feature in Complete Warrior, the duskblade class’s Arcane Channeling as well as the Smiting Spell metamagic feat in Player’s Handbook II, and so on.
It comes down to a DM ruling
Unfortunately, the rules aren't 100% clear on the issue of when exactly a reaction occurs, and Jeremy Crawford/Mike Mearls both seem to leave it to DM ruling for other reactions. (note that Crawford says "I have it happen after" as opposed to speaking in a rules-authoritative voice). The PHB errata also does not clear up this issue.
However, one reaction that occurs on being hit with an attack and that could potentially negate the attack is the Shield spell. Notably, it specifies that the +5 AC added can negate the triggering attack, and functions against other attacks until the start of your next turn.
Given that shield specifically calls this out, and other reactions-to-hit like Wrath of the Storm don't, you might infer that the hit generally takes place unless some specific rule negates it, and there is no specification in Wrath of the Storm. Other effects that can negate attacks like the Protection fighting style use the word "attack" instead of "hit". So in this case, you still get hit by the attack and suffer any consequences of the attack. Those consequences include taking damage and not being able to take reactions. Then the Wrath of the Storm reaction trigger occurs (being hit with an attack), but you can't use it. This effectively eliminates a corner case where, otherwise, you would get hit, but Wrath of the Storm kills the target before you take damage/suffer effects of the attack.
However, a DM may still rule otherwise, and it would be difficult to make a RAW argument opposing it (provided your DM was interested in hearing such an argument in the first place).
Best Answer
Yes, to both of your questions.
I can see why you'd be confused, since typically you can't hold the charge on a spell if you cast another spell. However, elemental touch isn't actually a touch spell. It's a personal spell that targets you. By casting elemental touch, you are giving yourself the ability to make a special melee touch attack for the next few rounds. If you cast shocking grasp afterwards, then you can make your elemental touch attack and have it discharge your shocking grasp as well. In addition, there's no reason that you wouldn't be able to use lightning for both. There are no rules about damage of the same type from multiple sources not stacking.
Be careful, though, that you cast elemental touch first. If you cast shocking grasp first, then casting elemental touch will discharge your shocking grasp to no effect, since casting a spell while you hold the charge discharges your touch spell.