why didn’t they kill Ellie when she said she was infected?

Because David doesn’t want to give her up that easily. In the game the man being interrogated by Joel even calls her “David’s newest pet.”

how did Ellie get out at the end when the doors were locked?

They are a fire escape which can’t be locked, save blocking or chaining them together. The front doors are locked but David missed the fire exit.

how the hell did Joel manage to capture and restrain two people and make a 2 mile trek in his condition??

Video game logic that doesn’t translate to TV very well. Ellie’s escape is also considerably longer in the game - sneaking about in the blizzard for a long time before reaching the restaurant.