If the two Pokemon you placed in the Daycare have an egg, the next time you run by the old man will call out to you.

As for the Breeding Mechanic, it follows all the same from Gen IV though there are two things to keep in mind. 1. It uses the IV inheriting mechanic used in HeartGold & SoulSilver and 2. If the mother has a Hidden Ability there is a 40% chance the Baby Pokemon will hatch with a Hidden Ability.

More about Pokemon Breeding can be read HERE.

1. Yes.

3. It's the same as Double Battles. Just with three Pokemon instead of two

4. Because

6. No, it won't be affected...not that it can have the ability

1. Yes, both of them will paralyze Ground-types. Body Slam is a Normal-type move so it'll hit Ground-types and though the Ability Static is connected with Electric-type Pokemon it still affects Ground-types if they physically hit or get hit for a Pokemon which has it.

2. There is a mechanic known as "priority" which organizes what action occurs in a battle in what order. While many moves priority are set at 0, there are some moves which have a higher priority then others. "Fake Out" has a +3 priority meaning many times it'll be used first, BUT "Helping Hand" has a +5 priority so even if the Pokemon you're battling is slower then you Helping Hand will always go first unless you do an action which has a higher priority of it. You can check out the list of what actions have what priorities HERE.

3. Triple Battles work just like Double Battles however they differ in the following ways:
* Your first three Pokemon in your party are sent out. The first Pokemon in your party is on the right, your second is in the middle, and third is on the left.
* Usually a Pokemon can only hit the Pokemon which are adjacent to it. This means Pokemon on the way left cannot hit Pokemon all the way to the right and Pokemon on the way right cannot hit Pokemon all the way on the left. Even if you knocked out your and your opponent's middle Pokemon the Pokemon on the way sides cannot hit a Pokemon on the other side. As always, though, there are moves which lets you hit any opponent or hit every Pokemon on the field.
If you want to find out more, such as what moves have special usage in a Triple Battle, go HERE.

4. There are many Pokemon which are a standalone, meaning don't have an evolution/pre-evolution, and it's the choice of the creators to have the Pokemon be like that. Also honestly I would actually think if these two Pokemon were going to get anything evolution wise it'll probably be a pre-evolution, though now that they have moves on from the Generation I - IV Pokemon I doubt we'll be seeing any evolution/pre-evolution for any of them anytime soon.

5. They won't get damaged as those Pokemon are set to receive 0x damage from Ground-type attacks. If a Ground-type used Earthquake on a Pokemon using Dig the damage normally done to the Pokemon is multiplied by 2x. However 0 x 2 is still 0, so thus a Pokemon with Levitate will still not receive any damage.

6. No, in fact if it had Flash Fire it'll actually be IMMUNE to Fire-type attacks (thus would make it have no weaknesses).

Really, I thought Bulbapedia described it well enough, but I'll give my go at it.

Unlike Triple Battles which are like Double Battles except with three Pokemon and adding in position strategy, Rotation Battles actually work much like Single Battles except in a few critical ways:

1. While only two Pokemon battle at any given time, you can rotate to the other Pokemon you have out. This is sort of like switching out, but this doesn't take a turn up meaning you can rotate and then attack with that Pokemon. Though be weary when you do this as your opponent can also decide to rotate and you won't know if they did so until after you declare an attack, so you must keep in mind all of your opponent's Pokemon.
2. Because the Pokemon is rotated out of action and not switched out, anything which has happened to the rotated out Pokemon is not reset. If a rotated out Pokemon was inflicted by a status ailment normally cured by switching out like confusion of badly poison (turning it into normal poison), had an Ability's effect happened like Slow Start finally de-activating, or had its stats decreased/increased in someway won't be reset and will have these effects activate once more if the Pokemon is rotated back in (though if a Pokemon is poisoned/badly poisoned/burned I don't think they receive damage each turn though I could be wrong). However a good thing about this is the rotated out Pokemon aren't affected by the attack Pursuit or the Abilities Shadow Tag and Arena Trap so you can switch them out if you want without fear (though doing so will take up your turn leaving your current rotated in Pokemon vulnerable to attack). Rotating out also doesn't activate moves or Abilities which involve switching out, such as Pursuit doing double damage or Natural Cure activating. Finally Abilities will only be active if the Pokemon is the one currently rotated in.
3. That's all I really know, I'm not to sure if you're able to apply items to your rotated out Pokemon. Honestly someone with Black Version would probably have more answers as that game specializes in that new battle mechanic (Pokemon White specializes in normal Triple Battles).

If this is still too much, think of it as this way: Rotation Battles are just Single Battles except any benefits/disadvantages you get from switching out a Pokemon doesn't occur if you rotate. Also Rotated out Pokemon don't have their Abilities activated and can be switched out safely and not be attacked by their opponents/stopped from being switched out (though a Ability which they have which involves switching out won't be activated due to rotated out Pokemon's Abilities being deactivated).

I don't think so.

If anything it helped Ash defeat Brock using Electric-type move. If you noticed during most early episode Pikachu was "charge" in someway (got hit with a lightning bolt to Thundershock the Spearow swarm away, got charged by a batch of Pikachu to blast Team Rocket off, etc.). You could say that all that charging which happened to Pikachu in that short of time might have increased it's Electric-type attacks.

However others might say because the early seasons weren't really structured they made Pikachu stronger for plot reasons and by the time they began "stabilizing" Pokemon Pikachu was technically high-leveled.