Remember that voltage is relative. The original supply had -12V, -5V, 0V, +5V, and +12V. When you re-purpose the supply to get 24 volts from the -12V and +12V lines, essentially you re-assign your "ground" or 0V to the -12V line. So now you have 0V, +7V, +12V, +17V, and +24V. Nothing is really different, you just changed your point of reference, ie. "ground." So the input of the 7805 will be +12V above your new ground and the output of the 7805 will be +5V above your new ground.
I don't think the heatsink will be necessary, the panel meter probably doesn't draw that much current.