Bonuses of the same type don't stack, so it won't do anything for your +3 battleaxe, since the +3 is an enhancement bonus.
(Bonuses that have no type at all - i.e. a +2 bonus - stack with everything.)
EDIT: Note that there is no difference between bonuses based on the source. If you are making a roll or calculating a value, bonuses of type X don't ever stack, no matter what they are attached to.
1. We get no benefit from the DIS feat if we cast with the off-hand implement.
2. We can cast through our off-hand implement...
...though we must designate main-hand and off-hand to determine the application of the feat as per above.
BUT: This is a weird fringe case with no explicit rules.
I'm gonna get wordy here, because so far as I can tell there is no official ruling on handedness as it relates to implements. The best we can do is look at weapon rules for these cases, and try to extrapolate, and then decide if that inference is valid.
Why am I talking about weapons?
Because Dual Implement Spellcasting did it first, and they're fundamentally the same thing for purposes of balance.
At their base mechanical function, weapons and implements serve identical purposes. They both provide a progressive enhancement bonus (+1 through +6) to the attack and damage rolls of powers channelled through them (and follow the same price advancement). With some bizarre exceptions they must both be held in at least one hand to be used. They each usually have powers or features that can only be used with attacks that use the item in question.
The feat in question uses a term ("off-hand") that is otherwise exclusively reserved for weapon mechanics. I believe that if they meant something more vague like "other hand," they probably would've said that instead; this may be optimistic on my part, but I'm running with it because I prefer "not so much leaping to conclusions as building elaborate cantilevered structures so as to reach conclusions more efficiently for prolonged periods."
So, what are the mechanics for off-hand weapons?
Off-Hand [from the DDI Compendium, also verbatim in RC 270]
An off-hand weapon is light enough that the attacker can hold it and attack effectively with it while also holding a weapon in his or her main hand. An adventurer can’t attack with both weapons in the same turn, unless he or she has a power that allows such an attack, but he or she can attack with either weapon.
I also found a parenthetical that may help clarify things, in the Two-Blade Fighting Style ranger feature [PHB1 104]:
(Make sure to designate on your character sheet which weapon is main and which is off-hand.)
In addition to this parenthetical, there's a plethora of items and features and powers which treat handedness as permanent ("while you wield this weapon in your off hand" isn't "when you consider this weapon to be in your off hand"), and many powers and feats (including this one) are obviously balanced on that assumption.
Together these indicate that handedness is permanent: characters are not ambidextrous, nor can they choose to change which hand is their primary. "Off-hand" is a mechanical term that means something definable and permanent to a character. (It also strains credulity that all adventurers are perfectly ambidextrous, but I won't argue realism on a 4e rules question.) This is the crux of the issue.
Dual Implement Spellcasting grants no unusual handedness privileges or exceptions.
If we allow ourselves to generalize what we learned from the off-hand weapon rules, when we dual-wield implements we must signify which implement is in the main- and which in the off-hand.
The off-hand weapon rules I've co-opted for implements say we get to choose which implement to attack through without penalty. This is how I answered question two.
But, DIS only gives the damage bonus when attacking with the main hand. This is how I answered question one.
If we want to cast through the implement currently in our off-hand and gain the enhancement bonus from the implement currently in our main hand, we must switch implements to the other hands. Feats and items which allow us to do this efficiently are outside the scope of this already-lengthy answer, so I'll link to Bag of Blades: A Guide to Off-Hand Properties and Free Action Item-Swapping and RPG.SE's own (slightly outdated) What action type is switching two items between main hand and off-hand?.
[On a side note, "staff" and "quarterstaff" are two separate items: a "staff" is a one-handed implement which can be wielded as a quarterstaff weapon but not enchanted as one, and a "quarterstaff" cannot be used or enchanted as a "staff" unless an exception is explicitly made. I mention this because my research for this question brought up dozens of attempts to use "DIS" with a single staff.]
Best Answer
Some basic math, but first a rule
From compendium:
Staffs (and other implements that are weapons) can apply their enhancement bonuses to regular weapon attacks made through the weapon (but powers and properties tend to be tied to implement attack powers.)
Thus you get the +2 to hit and +2 to damage on both implement attacks and weapon attacks made with the Magic Staff +2.
Math example: Implement attacks using your Int would be Int Mod + half level + enhancement bonus. Damage-wise its usually Mod Dam + feat + enhancement. Assuming you are level 6: 5+3+2+? = your bonus to hit. Assuming you have no feats (like expertise feats or damage focus feats) thats a +10 to hit and a +7 to damage.