Sort:  
  1. Re1 Txd5+
  2. Kxd5 Td8+
  3. Kc6 Qd7+
  4. Kb7 -> ? (intesd of 4. Kc5?)
  1. Re1 Txd5+
  2. Kxd5 Td8+
  3. Nd6 Qxd6+
  4. Ke4 -> ? (instead of Kc4?)

Good catches. Let me think here...

  1. Re1 Txd5+
  2. Kxd5 Td8+
  3. Kc6 Qd7+
  4. Kb7 Qb5
  5. Ka7 Qb6 checkmate
  1. Re1 Txd5+
  2. Kxd5 Td8+
  3. Nd6 Qxd6+
  4. Ke4 Qd4
  5. Kf5 Rd5 checkmate
    or
  6. Kf3.... I'm stumped here. Any move seems to open up too many possibilities. Perhaps someone smarter than me can solve it.
  1. Re1 Txd5+
  2. Kxd5 Td8+
  3. Kc6 Qd7+
  4. Kb7 Qb5+
  5. Ka7 Qb6+
  6. Sxb6 :-)

  1. Re1 Txd5+
  2. Kxd5 Td8+
  3. Nd6 Qxd6+
  4. Ke4 Qd4+
  5. Kf3 and white wins. :)

You are close, but ...

Qxd6 I think is a mistake, White doesn't have to move Kc4, he also has Ke4 now because black's queen is no longer guarding the e-file. You'll end up chasing the king to the wrong side of the board giving white a chance to defend w/ his other pieces.