Easiest heroic? Easiest badges?

Dependant on how many times you've run Durnholde while leveling, you should hit revered with KoT quite quickly. There's a few quests there that really boosts the factiongrind.

I was in the same situation as you with Heroics a few weeks back, and found SP a great place to start out. All of the bosses ar Tank n Spank and there's few or no really tricky pulls. (even though those dual defender hit hard for a poorly equipped tank. Just beware of it and have someone kite it if possible.)

HC Mech is also easy badges once ypu get the bosses right. A bit more strategy to this, so I'd suggest you read up on them (wowwiki for instance) before you start.

As for easy badges, nothing beats Bot when U can AoE it.

For someone undergeared Sp / Mechanar is definitively the way to go, bring a lock + rogue or other interrupter for Mech vil make it a lot easier. If you have have trouble on the Coilfang Defenders in Slave Pens bring a someone that can kite one of them or a rogue to stun lock.

If you bring 2 CC Steamvaults is also an option since nothing in there hits very hard (Except for the Bog Lords which are skippable anyway). So if you have enough CC to counter the Fear+Silence in some of the packs it becomes trivial.

When you start getting the gear to aoe tank instances UB, SH, Ramparts (& BM if you know some good dps'ers) becomes the fastest and easiest way to get badges. They really not the best place to start though.

Still, you should expect quite a few wipes until you learn the pulls in heroic mode. My first heroic was BF (without a lock) and was a horrible wipefest, once I learned the pulls it got rather trivial though.
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if you are familiar with slave pens from normal in your current gear it shouldnt be bad. You kinda need to know the place though, with really good dps I was clearin the place in 40 mins for 3 badges in my early blues.

you need to get the stuff from Caverns of Time rep it has served me tremendously.
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I haven't slept for ten days, because that would be too long.

Yeah, Keepers of Time rep was one of the first things I went after. The Continuum Blade has made a big difference. Also, +40 spelldamage to the blade, can't go wrong with more Spelldamage

As for the Badges of Justice being in the Reward Packages, I had 4 packs with a badge inside, 2 on Sunday, 1 on Monday and another on Wed. so, they are in them, it is just a matter of when they feel like giving them out.
Maybe I was a good little Paladin those days? Meh.
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I am the arm of the Light in battle. I know my path, can you say the same?

http://www.wowarmory.com/character-sheet.xml?r=Greymane&n=Bettiepage

the easiest badges is without Botanica. mobs there hit like pansies. now I have t4 + some 2.3 badge epics, but when I was decked in blues it was a breeze. and its a very fast instance if you're a good puller and your group isnt slackers/scrubs.

go Botanica.

Did Durnholde Heroic today, first time since 2.4 is on air.
peanuts.
Another 3 badges in 40 minutes.
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Cearn - 70 pew pew Spankadin! - EU Anetheron

It dont matter where you from its where you at~

Pally tank + good healer + heroic shattered halls = fastest 5 badges I've ever earned.
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What are the easiest?

I agree with all those that say Slave Pens and Mech are your best bet.

What are my favorite?

Well since you asked I like Shattered Halls and Mech. Slave Pens because it's easy and I've done it so much. Shattered Halls because if ever there was one it is an instance built for a pally tank.
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armory | youtube | stuff

aggro wipe on completion of sacrifice.
CC and interrupts, yep, setup is not wayne.
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Cearn - 70 pew pew Spankadin! - EU Anetheron

It dont matter where you from its where you at~