Induction - why do we allow $k=1$ in the second step? Having proved that the property holds for the first case (most often 1), in the second step we need to assume that it's true for some $k \ge 1$. Why not $k>1$

It's because if you don't you loose the connection with the first case. The idea is that you should in principle be able to step from the first case to any integer.
If you don't assume it's only valid for some $k>1$ and prove that it's valid for $k+1$ then that step would only be valid if you step from $2$ and higher. 
For counter example where this goes wrong we could consider the statement $P(n): n\ne 2$. First since $1\ne2$ we have that $P(1)$ is true. Now assume that $P(k)$ is true for some $k>1$ then $k+1>2$ so $k+1\ne 2$. This would mean that all positive integers are different from $2$ (which obviously isn't true).
