Why did Donnie blame himself for what happened? Did he cause Leo to run off? Can you be a bit more specific?
Sure, the differing of opinions the description states involves the point of conflict. Donnie's plan is to use the Turtle Mech, but Leo strongly disagrees and thinks the best plan is to make a second base outside the city. Meaning if Donnie had followed Leo's advice, they could have gotten out of the city with no invasion ever taking place. Leo gets separated from the others after a Kraang portal opens up between them blocking them from each other while escaping the lair, Leo tells them to get out while he distracts them to buy Mikey and April time to help Donnie who gets injured during the escape.

How did the subject of April even come up for Raph to even tell Casey that Donnie had a thing for her?
In typical Casey fashion, he just brings it up to Raph while spray painting himself with her on a building on the surface. Casey took the comment about Donnie and just was like "Pfft. Whatever man..."

You said April chose Casey. Mind specifying?
They don't really come out and say it, but the vibe is made pretty clear. When April is tending to Donnie's injury in her apartment, Donnie was basically like "Well, since it's the end of the world...again. I guess I can finally tell you how I feel about you April..." April gets wide eyed and then tightens the bandage around his arm to shut him up. Plus several others times it looked like he wanted to tell her after this, but she quickly changed the subject to the matter at hand, also the final scene in the Party Wagon, the embrace April does to Casey when she learns that Casey couldn't find his family looks very telling. Donnie's reaction doesn't look like he's going to take it well either...

How did Raph and Casey get separated from the others?
Raph and Casey start out separated from the others as they are on search for Karai.

What happened to the mech?
The Mech is fried as a result of their last ditch attempt to save the city.

What went down in the Shredder and Leatherhead encounter. Why were they going up against each other? Was he helping Leo?
Leatherhead actually saves Splinter from Shredder, the first fight that is. LH and Splinter meet each other in the sewers and head up to find the turtles, but do not. Sadly, Leatherhead's advanced age lead him to be defeated by Shredder and he looks like he falls through a floor of a demoed sight and into a running sewer. He isn't seen again after it.

How was Leo cornered?
Leo after being separated from the others is lured by the Kraang into the Foot's area to attack him in an all-out assault.

What was going on with tea time girl and pizza delivery boy?
They are really only seen for a brief period, basically the scene we see in the trailer.

Did Casey flee the city with the turtles? If that storyboard is any indication, and the trailer, I'm sure April did.
Yes both are with the turtles going to Northhampton. And all turtles too obviously...

How did Kirby get mutated? Was he protecting April?
Not really protecting April persay, he gets blindsided by Kraang Prime's mutagen gun weapon while trying to get him to "his" Party Wagon. The party is Kirby's vehicle when he was younger it appears.

Thanks. I should've rephrased the Leo question, though. How was he lured? The Kraang bots basically battled with Leo until he was drawn up to the surface where the foot took over.

Did they show Leo or Donnie's rooms? No they weren't, and sadly it doesn't look like they're going to be seen as the lair is pretty much burned and destroyed.