Learn Portuguese (Portugal): Estou mais ou menos bem.

Yes! Just like you might use muito (very) or pouco (a little) to describe how you feel, mais ou menos acts as a fixed phrase to express degree. It modifies bem (well) to mean you feel "so-so" or "more or less well".

Since we are learning about invariable words, does mais ou menos ever change its spelling?

No. When used as adverbs of degree, both mais (more) and menos (less) are invariable. This means they never change their endings to match masculine, feminine, singular, or plural words. It is always exactly mais ou menos.

Why do we use Estou instead of Sou for "I am" here?

We use the verb estar (which gives us estou) for temporary states, like how you are feeling right now or your current health. The other verb for "to be", ser (which gives us sou), is used for permanent traits.

Can I drop bem and just say Estou mais ou menos?

Yes, absolutely! In casual conversation, if someone asks how you are doing, you can simply answer Mais ou menos (So-so) all by itself or Estou mais ou menos.