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I was interested in knowing if all launch pads possess a flame deflector. I'm inclined to say yes, as the structure seems vital for a safe launch. Wanted to make sure, though, if there could be a case (for smaller rockets or some other reason) that would make it unnecessary.

Also, I know they have quite an impact in the produced sound field. Are they built, also, with the acoustics in mind, to reduce or shift the noise generation somehow? Or is its main purpose just to deflect the exhaust plume and the alleviate the heat?

Jak
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Welcome to the site. Flame deflections are not always used on small rockets

But as rockets become larger they are increasingly necessary at least on Earth, Flame deflectors such as flame trenches serve a number of purposes. Firstly they prevent damage to the rocket caused by the rebounding of foreign objects such as concrete particles up into the rocket. Secondly they ensure that hot exhaust gases are directed away from the launch pad facilities to minimise exposure.

Sound suppression is also important and the suppression is greatly enhanced by the use of large quantities of water sprayed directly below the rocket and into the flame trench. This also limits the erosion of the concrete by the hot exhaust gases.

There has been some discussion over the SpaceX Starship proposed landing on Mars and the need for a flame deflector for take-off from the Martian surface. The need for such a flame deflector has not been fully established but it may be necessary to build some form of flame deflector. The situation is complicated due to the low atmospheric pressure, low gravity, the unknown nature of the final landing site and the changing specification of Starship itself.

Slarty
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    As an interesting side note: Launch sites at the cape were built on land with a high water table so they couldn't excavate flame trenches. So instead they built up earth to a hill that the rocket sits on. So in this case the trenches are actually at ground level. The huge crawler that carries the rocket has pivot that keeps the rocket level as it goes up the ramp. – Johnny Robinson Mar 07 '19 at 05:07