I have done part a). I attached it incase it was useful in calculating part b.
For b) I am confused because vector product is usually for vectors.
I tried writing $\sigma=(x,y+rn \cos v ,z+rb \sin v)$
Then calculated the 2 derivatives but I do not get the required answer. Maybe I need to understand what the vector should be first.
$\newcommand{\Vec}[1]{\mathbf{#1}}\newcommand{\Tgt}{\Vec{t}}\newcommand{\Nml}{\Vec{n}}\newcommand{\Bin}{\Vec{b}}\newcommand{\NML}{\Vec{N}}$Since $\gamma$ is a unit-speed parametrization, $\gamma'(s) = \Tgt$. The Frenet-Serret equations read \begin{alignat*}{3} \Tgt_{s} &= &&\quad\kappa \Nml, && \\ \Nml_{s} &= -\kappa \Tgt && &&+ \tau \Bin, \\ \Bin_{s} &= &&-\tau \Nml. && \end{alignat*} By definition, $$ \sigma(s, v) = \gamma(s) + r(\Nml \cos v + \Bin \sin v), $$ so the Frenet-Serret equations give \begin{align*} \sigma_{v} &= r(-\Nml\sin v + \Bin\cos v), \\ \sigma_{s} &= \Tgt + r(\Nml_{s}\cos v + \Bin_{s}\sin v) \\ &= \Tgt + r(-\Tgt\kappa\cos v + \Bin\tau\cos v - \Nml\tau\sin v) \\ &= \Tgt(1 - r\kappa\cos v) + \tau\sigma_{v}. \end{align*} Since $\Tgt \times \Nml = \Bin$, $\Nml \times \Bin = \Tgt$, and $\Bin \times \Tgt = \Nml$, \begin{align*} \sigma_{s} \times \sigma_{v} &= \bigl[\Tgt(1 - r\kappa\cos v) + \tau\sigma_{v}\bigr] \times \sigma_{v} = \Tgt(1 - r\kappa\cos v) \times \sigma_{v} \\ &= \Tgt(1 - r\kappa\cos v) \times r(-\Nml\sin v + \Bin\cos v) \\ &= r(1 - r\kappa\cos v)(-\Bin\sin v - \Nml\cos v). \end{align*} As in the comments, $\NML = -\Bin\sin v - \Nml\cos v$ is a unit normal field to your surface, and $$ \|\sigma_{s} \times \sigma_{v}\| = r(1 - r\kappa\cos v). $$
Now, \begin{align*} \NML_{v} &= -\Bin\cos v + \Nml\sin v, \\ \NML_{s} &= \Nml\tau\sin v - (-\kappa\Tgt + \tau\Bin)\cos v = \Tgt\kappa\cos v + \tau\NML_{v}, \\ \NML_{s} \times \NML_{v} &= \Tgt\kappa\cos v \times (-\Bin\cos v + \Nml\sin v) \\ &= \kappa\cos v(\Nml\cos v + \Bin\sin v) = K\, \sigma_{s} \times \sigma_{v}, \end{align*} from which you can extract the Gaussian curvature $K$.