I know figs cannont end movement in blocking terrain, but can WoS Nightcrawler phase into the center of the middle of a blocking terrain area as shown below, and (if perplexed) break the square of terrain?

X X X X
X T X X
X X X X
X X X X

X = Blocking Terrain
T = Target square.

The strategy being that once he breaks that square he can sit inside and use it as a base of operations. Phasing out, attacking, and returning.

No, because when using hypersonic speed, you have to make your attack from a square that you can legally end your movement in. Nightcrawler can't legally end his movement in the square with blocking terrain, so he can't attack from it either.

The Rule of Occupancy says "heck no!"
When the attack is made, Nightcrawler cannot end his movement in the square he occupies.

Next question. Say Nightcrawler is carrying someone into a cell on the jail map and is making an attack at the same time. There is only one free cell in the square and it is in the corner surrounded by blocking. Can he move into that square since the carried person in a HSS is considered "off the board"?

You need to make the attack from a position where you could legally end your move. If you cannot put down your passenger you cannot make the attack.