Speech itself is also covered by freedom of speech, does that mean you can shout into your neighbours property for hours a day? With amplification? From a recording?

No, but that’s often regulated against, and much easier politically to deal with.

The problem is that a flag is, well, a flag. It’s treated as if they’d planted a tree and I’m upset but the wind rustling the leaves.

I’ve looked for anything analogous where wind acts upon an object to make loud noises, and I haven’t had luck.

I actually thought the flag code disallowed flying the flag in bad weather. But I'm now unable to find any confirmation of that.