FM-PAK producing a lot of audio noise (hum/buzz)

tl074 is audio wise the best sounding. Tons of 2th order harmonics and nothing else. While the specs are much worse then other opamps, harmonic wise this is simply the best. As a added bonus the signal is very stable as its too slow to start oscilating in the inaudible spectrum.

Would it be wise to have a ground plane on the PCB?

Those are not ground planes. A ground plane is a continuous in all directions plane with no interruptions. Otherwise you just have very wide ground traces that may or may not be any better than just regular old thin ground traces going wherever, depending entirely on how they're routed.

Signals couple to the nearest ground that's available, and if it's not so near, you'll end up with a lot of EMI (electro-magnetic interference) being emitted. Here's an example of a routing that will produce a lot of EMI, where = is a ground trace and - is a signal trace:

      +---------------------+
      |                     |
    --+                     +--
    ===========================
    ===========================
    ===========================

The thickness of that ground trace is irrelevant; you're going to get a lot of EMI anyway. The better solution would be to use a thin ground trace (as thin as the signal trace) that follows the signal trace with as little distance between them as possible.:

      +---------------------+
      |+===================+|
    --+|                   |+--
    ===+                   +===

Here's a good picture, from this video of what happens when you route signal lines on one side of the board across an area of the board without ground on the other side, even though you have otherwise a very thick ground plane:

The thin black lines are the signal path, and the red and yellow parts are the ground return path. You'll notice that the signals and the ground return get really far apart over the gap; the electric field that's carrying the energy will extend between the two, affecting everything in between them. (I.e., signals on the bottom of the three signal traces in the gap will be coupling into the two above, introducing noise.)

The TLDR here is, you don't need a ground plane: you need ground traces following as closely as possible your signal traces. A real ground plane (covering the entire side or layer of a board) is just an easy way to do that since you don't have to worry about routing ground traces: any signal trace you route will always be your board thickness away from ground.

I have some more details about this in the Signal Return Paths section of of one of my notes. As well as referencing the above, it also references Eric Bogatin's brilliant presentation The Value of the White Space.

Principle number 3 from his "Top Ten Signal Integrity Principles" in Signal and Power Integrity—Simplified is a really good way of thinking about it:

Well, I'm not so familiar with Vcc planes since I don't do four-layer boards, but I'd imagine it mainly just makes getting VCC to anywhere you need it much easier, since you can just bring a via up from the Vcc plane anywhere. It can also, depending on the thickness of the material between it and the ground plane, help create a decoupling capacitor between Vcc and ground that's "everywhere", but IIRC the effect was minimal to nonexistent if you did the typical Vcc/GND on layers 2 and 3, which are much more widely separated than layers 1 and 2 and layers 3 and 4.

But if you go back to the key insight: that every signal path couples to the nearest ground for its return path (technically, the return path offering the lowest impedance), and that the energy is transferred not through the traces but through the _field between the traces,_ you'll see that any noise on power lines will also couple to the nearest ground, so anywhere a power line might be noisy and the noise may be problem, you want a good return path for that, too.

(I should mention, I'm not an EE, nor do I even play one on TV. If an actual real EE gives you advice, and is providing explanations more or less consistent with the above (or pointing out the problems with it), you're probably better off following him. But it's also worth keeping in mind that a fair number of EEs don't actually well understand this stuff, and that was probably most EEs back in the '80s and '90s.

BTW, it makes threads easier to read if you edit down quotes to just the particular part your responding to, rather than making people scroll through the whole long post a second time. Or even just leave off quoting completely, if it's just a general reply that doesn't reference any particular bits specifically in the previous post.