If my strings skill goes from 20 to 40 will my heal song heal more hp's with a lute equiped or what?
My take:
No, skill ups will not improve the songs effectiveness. It affects only missed notes.
I had a low-level bard (~12) tell me once that he ran faster when he got drum skill-ups. But "faster" is pretty subjective, and of course he didn't do any actual measurements.
I've never seen any evidence that skill-ups affect the song's effectiveness. It's always been the case that if you look at the song stats on lucy.allakhazam.com, and then multiply the stat, etc. by the song's (instrument modifier / 10), you'll get the actual stat value increase without regard to instrument skill.
For example, if you use a lute with a 21 modifier, you'll get 2.1 times the normal healing effect with the lute equipped.
There are many songs that specify an instrument to be used, but where the instrument doesn't actually affect the song in any way. Mezz is one of those, though I have had people tell me that they get a lower resist rate when using a flute. I've never attempted to verify this. The enduring breath, invis, and the levitation songs are other examples.