The period of oscillation of a mass-spring system is given by:
$$\boxed{T=2\pi \sqrt{\frac{m}{k}}}$$
So the weight of the bob $m\mathbf{g}$ has no effect on it, only its actual mass $m$.
It would be different if it was a swinging pendulum.
To answer the central question the OP asked, we need to solve the Newtonian equation of motion. Armed with that knowledge we'll be able to calculate the maximum velocity.
So, consider the Newtonian equation of motion for a mass oscillating in the $y$-direction:
$$F_{net}=ma_y=m\frac{\text{d}^2y}{\text{d}t^2}=-ky+mg$$
$$my''(t)+ky(t)-mg=0\tag{1}$$
Substitute:
$$u=ky-mg$$
so:
$$u'=ky'$$
and:
$$u''=ky''\Rightarrow y''=\frac{u''}{k}$$
Insert into $(1)$:
$$\frac{mu''}{k}+u=0\Rightarrow u''+\frac{k}{m}u=0$$
Now substitute:
$$\omega^2=\frac{k}{m}$$
So that:
$$u''(t)+\omega^2u(t)=0$$
Which is the classical diff. equation of a harmonic oscillator and has the solution:
$$u(t)=A\cos(\omega t+\varphi)$$
Where $A$ is the amplitude and $\varphi$ the phase angle.
The initial conditions are:
$$y(0)=A\text{ and }y'(0)=0$$
Back-substituting we get:
$$y(t)=\frac{1}{k}\Big(A\cos(\omega t+\varphi)-mg\Big)$$
Using the first initial condition we can now show that:
$$A=\frac{1}{k}(A\cos\varphi-mg)$$
$$kA=A\cos\varphi-mg$$
$$(k-\cos\varphi)A=-mg$$
$$A=\frac{-mg}{k-\cos\varphi}$$
So we can write:
$$y(t)=-\frac{mg}{k(k-\cos\varphi)}\cos(\omega t+\varphi)-\frac{mg}{k}$$
(We don't need to determine the value of $\varphi$ here)
The velocity (here as absolute value) is given by:
$$|y'(t)|=|\frac{\text{d}y(t)}{\text{d}t}|$$
Which gives:
$$|y'(t)|=|\frac{mg\omega}{k(k-\cos\varphi)}\sin(\omega t+\varphi)|$$
This reaches a maximum for $(\omega t+\varphi)=n\frac{\pi}{2}$ with $n=1,3,5,...$ because then the $\sin$ becomes $1$.
Or:
$$\boxed{|y'(t)|_{max}=\frac{mg\omega}{k(k-\cos\varphi)}}$$
So the maximum velocity does indeed depend on $g$.
In a simple sense this is quite easy to understand. For a generic oscillator:
$$y(t)=A\cos(\omega t+\varphi)$$
the maximum velocity is in fact $\omega A$. It so happens that in our case $A$ is dependent on $g$.
