The cabinet does look well made. Jims suggestion of screwing and gluing reinforcing strips of hard wood front to back is a good one, though probably not necessary. If I wanted to reinforce the bottom shelf I would make my rip about a half an inch less than it would take to touch the ground (like the base board across the front and sides) and run it long ways down the center of the cabinet. You could attach it using glue/nailer blocks like the ones that they used during original construction (the 1" x 1" (ish) wood strips with screws through them, along the front and sides). Just make sure you use the right length screws so that they don't break through the top surface. The glue is the important part here, that is what makes the permanent joint. The screws just apply pressure (also important) until the glue dries. My guess is that you will be fine without any additional structure though.
Ventilation without the back should be more than enough, however I would strongly suggest that you put a piece of 1/4" sheeting (or cut down the original back) where the drawers are. The back is what gives you your "sheer" strength. It may feel sturdy right now, but once you try to move it with some weight in there, or someone leans on it or sits on it and shifts their weight... it's suprisingly easy to snap every joint with a good sharp hit directly to the corner of a backless cabinet, and once it goes... it's gone. Yours is well made, and gusseted with good industrial glue, but there are no mortise and tenon joints, no dados... just straight forward butt joints (easiest to make, but also the weakest). That back, flimsy as it may seem is an important structural element to the cabinets integrity.
Hope that helps.
Cheers and good luck,
Greg
edit: just noticed that the doors are beveled glass. You should leave them open when running your equipment, or if you prefer them closed then you might want to invest in a small, silent computer fan to circulate the air around your amp.