You could just solve your first equation for $T$....

Simplistically:

For a circular orbit orbital velocity is constant at $$\sqrt{(G/r)(M+m)}$$

So for orbital period: you know $r$ (also constant), so calculate the length of the orbit (circumference, assuming it's a circle) and period is just length / velocity

Notice for a given primary the only thing that really matters is $r$.

Mass of the satellite (assuming it's artificial) is kinda negligible ($M+m$) where $M$ is the primary.

You are right about the geosynchronous orbit...but the period is determined by the radius...

props to HDE 22686 for adding the math graphics.