Cruzing - good job and thanks for your post with pics.
You have obviously chosen to use line voltage and high quality industrial parts. I especially like your attention to details like the muffler on the flow controller (min pressure valve) and liquid filled gauge.
How did you mount the timer - to what and how is it attached? I see the socket and the module. For people who may not be familiar with the module used here, it is a repeat cycle timer that plugs into a standard socket for wiring. The modules are available in various ranges such as seconds and/or minutes.
What is the timer controlling? Is the timer just controlling the relief valve? If so, is your pump running all the time?
A schematic might be helpful to anyone wanting to build a similar system. Also helpful would be a parts and vendor list. You have already done a great job by taking time to post pics!
Unfortunately, I don’t have any solutions to the sound problem other than using a box lined with foam over the entire system. Cardboard and open-cell egg crate foam (mattress topper) - use spray adhesive to attach the foam to the inside of the box. However, heat build-up within the box can be a problem. Typically these vacuum pumps have thermal overload protection and when they get too hot they shut down. I suppose you could put a temperature probe in the box to alert you to high temps.
My latest pump project is awaiting parts from China - must be on a slow boat!