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When we install gun safes on concrete, we use 3/4" plywood cut 1/2" larger than the width and depth. This gives us 1/4" on each side. We drill from inside the safe, and use 3/8" or 1/2" sleeve anchors that are at least 3" long. These are available from most hardware stores.
The problem with having an air space beneath the safe is that it also creates a prying point. I would leave the carpet as it is, and install the bolts through it. You can use a exacto knife or gasket punch to accomplish this.
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