Is an exhausted PSU safe to use for an FPGA?
Can the PSU sustain all the modules?


I wouldn't count on it. Most of the power from a modern power supply is on the 12V rail, not 5V. A typical profile for a modern 300W power supply looks like this, for instance -- 16A at 5V is the relevant limit:



This one happens to barely be within your margins, but not enough to be comfortable. Given that you aren't certain about your power draw, and that the power supply you plan on using may have different specifications (and may be on its way out!), I wouldn't count on it. Get a dedicated 5V power supply if you need this kind of power -- take a look at Meanwell products, for instance.