Learn German: Wir hatten es sehr eilig und waren total schusselig. - We were in a huge hurry and were totally scatterbrained.

This is a fixed expression: es eilig haben means to be in a hurry. The es acts as a dummy object that does not translate to anything specific in English, but it is required for the phrase to work. You will see this mandatory es in several German idioms.

Could I say Wir hatten es viel eilig to mean we were in a huge hurry?

No, because eilig functions as an adjective or adverb here, not a noun. To intensify it, you must use sehr (very) or besonders (especially), rather than viel (much).

What exactly does schusselig mean?

schusselig is a common colloquial word meaning scatterbrained, forgetful, or somewhat clumsy in a frantic way. It perfectly describes the state of dropping things or misplacing your keys because you are rushing.

If I started the sentence with a time word like Gestern, where would the es go?

It would immediately follow the subject: Gestern hatten wir es sehr eilig. The dummy es generally stays as close to the conjugated verb and the subject as possible in these fixed phrases.