What do you mean by “xn=0” .  Xn is a random variable , it can take two possible value with different probabilities.

Prob(2)

As you want. Ineed, convergnce in probabilty to a constant is indeed equivalent to convergence in distribution

I think giving details about (b) is not a good choice although it is simple. convergence in probability implies convergence in distribution. this is a very common result, you can find this on wiki or your textbook. If you still want the details, i can write the proof of b to you.

Prob(1)

I updated the solution. If you have any question, please tell me where you do not understand.

Prob