The sentences that demonstrate the correct use of hyphens are:
a) The café I work for is family-owned.
d) I work for a very lively family-owned café.
Hyphens are used in compound adjectives to connect two or more words that work together to modify a noun. In these sentences, "family-owned" is a compound adjective describing the café, and "very lively" is a compound adjective describing the café as well.