1. For probability density use the born approximation formulae (psi)*.(PSI)dv. This is your probability density and as you haven't given any limits then either you have to find it in unit volume or you have to left it in the differential form. For unit volume the answer will be A^2 and for dv=r^2 sin(theta) dr dtheta dfi , you get A^2 multiplied by dv.
2. Now for the Probability current density , find (psi)* i.e., the complex conjugate of psi and substitute in the formulate of probability current. Take components of vec(r) & vec(k) in cartesian co-ordinate and the find the the value of grad(psi) and grad(psi)*. The probability current is given by : J= i.hcut/2m [ psi . grad(psi*)- psi*. grad(psi)]. Substituting the values you will get hcut (A^2/m) multiplied by  [k1+k2+k3] where k1 , k2 , k3 are the x , y ,z components of the vec(k).

I hope my answer helps you. Wish you all the very best for your future endeavors!