For the second part: First, none of those answers lines up with what is being asked, so none are correct. That said, if the question were asking what the velocity of the object was when potential energy is 2/3E, we could solve it as follows:
Total Energy E = PE + KE. So, if the object has a PE that is 2/3E, then it must have a KE = 1/3E.
As in the earlier part, Total Energy E is also 1/2(kA^2). We can set up two equations for KE.
KE=1/2mv^2
KE=1/3*(1/2*(kA^2))=1/6(kA^2)
We can then set those equations equal to each other and solve for v.
1/2(mv^2)=1/6(kA^2)
3(mv^2)=(kA^2)
(3m)v^2=(kA^2)
v^2=(kA^2)/(3m)
v^2=(k/m)*((A^2)/3)
v=+-(sqrt(k/m)*sqt((A^2)/3))=+-(sqrt(k/m)*(A/sqrt(3)))