No, County Judge is a title left over from the reconstruction days. It's a completely different office than that of a court room judge. Purely administrative function, over the county bureaucracy.
IIRC, he does have some judicial respnsibilities, but mostly over school truancy cases. I don’t think he actually hears cases.
Depends on the county. The "County Judge" is the chief executive officer of the county, a full voting member of the Commissioners Court, which is legislative body for the county, and the presiding judge of the Constitutional County Court. In very large counties (1 Million plus population I think) the County Judge may elect to surrender their judicial duties. If they do that they are not bound the Judicial Code of Conduct, and are as free to run their mouths about politics as any other non-judge politician.
In very large counties there are numerous County Courts at Law, and often Probate Courts too, all of which perform the duties of the Constitutional County Court, and there is no real need for the County Judge to perform judicial duties. But there are many, probably in excess of 100, counties with no County Court at Law, and the County Judge is the judicial officer who tries Class A & B Misdemeanors (up to 1 year in jail), and hears probate cases.
In those counties the County Judge not only sentences defendants to jail, he/she votes to set the county tax rate, and faces the heat for that. Making the judge take the political heat for the taxes required to staff and operate the jail tends to discourage "lock 'em up and throw away the key" thinking.