Fixing an LED Home Cinema display caused whistling and a spark, is it salvageable?
caused whistling and a spark, is it salvageable?  Does the whistling and the spark mean that one of the two boards is fried?

This sounds like you shorted something to ground.  The whistling is most likely from a blown capacitor.  I've intentionally shorted them for demo purposes and some did in fact whistle.  While it could be a different component, my bet is a blown cap.

the logic board whistled, had a spark, and produced smoke for a few seconds

This is a classic case of &quot;you let the magic smoke out.&quot;  In all seriousness, a spark and the sign/odor of smoke is a clear indication that you shorted something causing it to blow.
Now, there should be two boards - the power supply and the logic board.  It's entirely possible that the damage is to the logic board but the descriptors used point to the power supply.  Either way, without putting hands and eyes on it, we can only &quot;best guess&quot; it at this point.

It stayed in a slightly damp basement for one year: when I opened it apart, it had a bit of mold on the metal....although some of the metal on the power supply board had a layer of rust

Oh, oh.  This is a very good indicator that there are shorted connections.  Water may cause mold which in and of itself is easy to deal with, but moisture causes corrosion which is conductive.  Rust?  That too is conductive and should have at least been removed.
Can it be salvaged?
Technically yes.  But is it worth it?  I doubt it.
Logic boards and power supply boards can have components replaced quite easily. However, being that there is corrosion and rust with an unknown amount of time &quot;infecting&quot; the PCB; it's impossible to tell how much damage and how many layers are workable.
You may have to get replacements.  They are available on second hand markets like eBay.  (Apple no longer sells or supports these monitors).  However you're ultimately spending money and time on what amounts a 10 year old FHD monitor.  I have three in my garage right now (19 to 22&quot;) that I can't even give away.
This is probably something you want to sell as parts given the likely damage and the missing glass (which protects the LCD, prevents dust and contaminates from getting in, and enhances the colors).  There are simply too many better (new) monitors you can get for a few dollars more.