I know about the order of the sharps and flats, C major has no flats or sharps, A minor is the same, F and dm etc.

But what if the "solo" is Lydian or Dorian? Do the major and minor keys become pointless, or something to base it off of?

I know it's about the root note, but I can't really tell when there's a progression going.

Is the only way to find out, is to strum a chord or root note and try to find the scale over it starting on the root note?

Satriani's Extremist for example.
Read the sticky.
Someones knowledge of guitar companies spelling determines what amps you can own. Really smart people can own things like Framus because they sound like they might be spelled with a "y" but they aren't.