Will the water in the bucket be considered as Paak or Napaak?

We understand from your query that a bit of soap that was applied to one’s body, fell into the bucket whilst taking Ghusl. If that area of the body (where the soap was applied) was Paak, then the water (in the bucket) will also remain Paak and suitable to continue with the Ghusl. The soap will not become Napaak, nor can it be considered as Najaasat, due to applying it onto one’s body in Ghusl.

Yes, if the soaped area was Napaak from before (that is, the impurity wasn’t washed off before soaping the body) then it will cause the applied soap to become impure as well. If this soap now falls into the bucket water then the water (in the bucket) will become impure and will have to be changed before continuing with one’s Ghusl.