Since you have a lot of photos you want to copy you should to construct a copy photography production station. I worked at a photo lab back in the day and part of my work was doing just this. The setup requires a steady tripod to enable you shoot down on your photos; the tape-hinged glass cover/carrier; lintless gloves to handle the photos; canned air to blow dust off the underside of the glass, the photos themselves before placing them under the glass, and off the top surface of the glass; the two lights with polarizing filter sheet material over them at 45 degree angles to the photo carrier, a black card gobo with a circular hole cut in it the size of the front barrel of your lens and a polarizing filter for your camera lens to hold that gobo in place with the front of the lens poking through which eliminates any glare; and a place where you can set all this up and control the ambient light in the space (by turning it all off) so your working lights provide the only illumination in the space & a remote or cable release so you can fire your camera hands off.
Group all your photos by size so you're not constantly raising & lowering the camera or having to zoom in & out & refocus with every shot. Manual focus may be the way to go.
If you want the detailed step by step just say so...
You could construct a small vacuum table by building a flat box out of something like white melamine-coated particle board. Drill a number of very small holes (like 1/32") in a grid pattern in the top, with a larger hole in the side. Connect a small Shop-Vac or similar to the side, place a photo on top so that it covers most of the holes, and let atmospheric pressure do the rest. You might want to include an adjustable vent in the side of the box so that you can control the force of the vacuum.
Another possibility is to use a bit of repositionable spray adhesive like 3M 75. Spray a bit on a flat substrate and you'll be able to stick the photos to the backing temporarily. The photos should release pretty easily with little or no residue left on the photo. This method would be quieter, but if the originals are at all fragile or if they have any value, I'd pick the vacuum plan because it should have less chance of damaging the photos.
Stop down to smaller than at least f/8. The increased depth of field will mean you have increased sharpness / more acceptable focus for the elements of the picture that are above the work surface.
Paperweights on the corners / edges.
"Mount" the photo to a magnetic-responsive surface using refrigerator magnets (really, this is the same as the previous option, just using magnetic attraction instead of gravity as the field force).
Build a "reverse air hockey" table, with lots of little holes in the surface connected to a suction pump, that will suck the sheets down to the table.
In all seriousness though, unless you have better reasons than what you listed for not using the glass pane you have... well, I recommend the glass. The issues you listed are not reasons not to use the glass. They are reasons to improve your technique.
Fingerprints: use gloves. Or better yet, tape one edge of the glass to your worksurface, and on the opposite edge of the glass, create a tape flag or "handle" to allow you to lift the pane without touching the glass surface.
Reflections: use two lights, on either side of the photo, aimed at 45° to the glass. Use dark posterboard or cloth as gobos to block reflections from other light sources in the room.