Does changing the volume of a reactant affect the rate of reaction? Does changing the volume of HCl affect the rate of a Mg & HCl reaction?
More volume of HCL does not increase the concentration and therefore does not increase the likelihood and so rate of reaction. 
Mg and HCl forms a metal salt MgCl and produces Hydrogen gas.

There may be a very slight increase in rate of reaction but this is permissible/ very negligible due to the fact that metals and acids form adducts and these can further react/ delay further reactions between the acid and metal molecules.

The rate would definitely be affected by factors such as temperature of the (aq) HCL. This is until the reactant Mg is used up then the reaction will cease leaving you with a flat curve on a rate graph.

Now, for Higher concentrations- this implies that more reacting molecules are at higher proximity to each other therefore the intermolecular collisions between the Mg and HCl molecules in your case are more frequent therefore forming more product per unit time. 

Control variable: mass of Mg to at least 1.d.p, concentration of acid moldm-3, volume of HCl cm3 - even though this doesn't affect the rate of reaction enough - they would want you to keep this constant for valid test etc.

Hope this helps!