May the commissioners court place non-binding referenda on the ballot to gauge the public's opinion?
Generally not. Unlike home rule cities which can have procedures for initiative and referenda in their charter, there is no general authorization for a county to hold non-binding referenda, except with specific statutory authorization for the particular subject matter. For example, Local Government Code Section 291.007 allows a county with a population under 40,000 to hold a non-binding referendum on any matter affecting county property.