If the US president commits murder during his vacation as an unofficial action during his presidential term, can he be charged with a crime in court (i.e., in the judiciary branch) before he steps down as a president?
Yes. The absence of immunity for a U.S. President's unofficial acts was established both in the Nixon Administration and later in the Clinton Administration.
In practice, a prosecutor would be loath to file such charges absent very, very solid probable cause, and a court would often be very deferential in accommodating the President's schedule and, for example, in allowing appearances by telephone when allowed by law, or by electing not to seize the President's passport as a bail condition to be free pending trial, as would be common for someone facing felony charges pending trial.
But, ultimately, the President does not have the right to either defer the charges until the completion of his term, or to any immunity from charges for his or her unofficial acts.