Yes almost all SSD drives will tell you power requirements, though the currents are generally given as peak current, the operating current is normally lower than that, but it is for sizing of power demand. Almost all just are fine with 5V only, simply not connecting the 3V3 and 12V pins, or having a pull up on the 3V3 instead to provide voltage there if the drive uses the 3V3 as power on rail. internal to the SSD the 5V will be converted to an IO voltage of 3V or so, and a core voltage that depends on chipset, generally something from 0V9 to 2V0 to run the logic, and a separate supply on each flash die that generates the needed erase and write voltages, runnning off the internally made 3V3 or 5V supply. Most only need 5V, and often laptop spinning drives also only use 5V as well, not needing anything other than this single rail.
Note the adaptor you show only seems to use the 5V from the Molex socket, being designed as interface from older IDE to newer SATA, so it has to use the molex to get 5V supply. The link is probably set wrong, try with it not connected to any pin, which will probably let it work, the 2 regulators getting hot is likely the power demand for the on board ASIC, needing likely a 3V3 rail, and the power being too much for a single linear regulator, so there are 2 in parallel fed off the 5V rail.