Use of a DC power supply and an arbitrary wave generator in digital circuits
The waveform generator can make a variety of shapes: sine, triangle, sawtooth, square, or one of your own design (that's the &quot;arbitrary&quot; part.) It can certainly replace a 555 as a clock source.
To do that, your generator should have a setting to make logic pulses, that is square-waves that swing from 0 to some + logic voltage (e.g., 5V or 3.3V). Try this out by connecting your scope to the AWG, and experiment with the settings to get the desired waveform.
Finally, the AWG can in theory generate DC, but it's not a good idea to use it that way as a power supply as you risk damaging it. Your power supply is much cheaper and easier to fix or replace, and is much more robust if you make a mistake. Also note that the signal gen has an output impedance of 50 ohms, so the delivered DC isn't going to be what you expect because of the IR drop in that impedance.
In any event, the best policy with an AWG is to first set up and check it with a scope to see if it's doing what you expect, then connect it to your system. Assuming you have your signals right, connect the ground of the generator to your system ground, and signal to signal.
Similar things come up with your power supply. Set it up, and check it first before you connect it. You can measure it with a scope, but you may find it handier to use a multimeter.