Does search still work normally on fields with norms set to false?

I am looking at saving disk space for an ES index and read that setting a text field with "norms": false helps.

If a text field were to have norm = false, other than scoring, do other aspects of search still work e.g. phrase search, *, fuzzy, Boolean operators ?

Yep, full-text search and all the various query components should continue to work as normal. It'll just affect the scoring of those fields since they lack some of the normalization components.