Do dead guitar strings sound better?

Do rusty, musty and crusty strings lead to a better tone?

Answer: It can, depending on circumstances.

I'm usually pretty good at changing out strings regularly, but there are times when I just don't feel like doing it and the existing strings stay on for a while longer.

There is one thing about old guitar strings that makes them good, and it's this:

They're already stretched.

Guitar strings, at least of the traditional electric 3-unwound/3-wound variety (which is most of them,) get to a point where they stretch and keep really good tune. When a set of strings gets too old, then sure, they lose tune easily. But sometimes a set that's been on the guitar a while gets that "sweet spot" where you can play for hours and your axe just sings.

Is there a specific type of string that gets the "sweet spot" best when left on the guitar a while?

Yes. It's the string type you probably already use. Nickel-plated steel.

Whether it's D'Addario, Fender, Dunlop, GHS or whatever you use, the nickel-plated steel wears in a way that can really achieve that nice worn-in "mellow" feel. Heck, it even works for metal tones.

HOWEVER... I notice that strings with ball ends are the best kind. An example of a string that doesn't have a ball end is the Fender Super Bullets; they have bullet-shaped ends. I used to use those a lot but they kept going dead too quickly for my playing style. The ball ends just seem to work much better, and I get more choices of string brands to choose from.

"How do I test this for myself?"

Easy enough. Leave a set of strings on your guitar a while and see what happens.

But I have a few tips:

Wipe down the strings regularly to prevent gunk.
If you see string denting, the strings have been on too long. Switch them out.
If you see black marks, that's rust, and that's bad. Change strings if you see that happen. You don't want to be forcing rust into your finger's skin.
If you leave your guitar in the open on a stand, cover the neck to prevent dust build-up. A towel usually works best.
Try to bend strings less to make your existing set last longer. Chord more.

Best ZOOM R8 tutorial book
highly rated, get recording quick!