Generally yes. The reason is because sometimes juries have questions. If the jury has a question they write a note and have it sent to the court. The judge has to read the note in open court, on the record, and the parties must be present to discuss the question and debate how the judge should answer it. Then the jury comes back in and the answer is given to the jury. If the defendant wasn't at the courthouse then they will have to wait for the defendant to come back, causing major delays. Likewise, if the jury reaches a verdict then they wouldn't be able to come back to court and deliver it if the defendant wasn't there.
This is not to be considered legal advice nor does an attorney-client relationship exist.
Legally the answer is no. If you are out on bond then you are free to come and go (subject t the conditions of your bond) until a guilty verdict is entered.
Practically the answer is it depends. If the Judge orders you to remain in the courtroom then you must, if not see above. However, the Defendant (and the attorneys) would have to be close enough to be in the courtroom within a reasonable amount of time in the event of a jury question, verdict or other issue.
Yes, most times the Defendant is told to stay at the court house. The reason being is the jury may have a question, come back with a verdict, or adjourn for the day and the Defendant needs to be present for these. Judges do not like for juries to have to wait more then they need to so having the parties close is common practice. Depending on the case, there is a possibility the defendant may run and in order to prevent this, they are ordered to stay in the court room. When the verdict is read, the deputies always block the exits so the defendant can't run if they don't like the verdict. There have been numerous cases where defendants did not like how the trial went or are scared of going to jail so they run, thus why the court has the defendant stay in the court room if not the court house during the deliberations.
Some judges will insist that Defendant stays within the court room. Other judges will just insist that the Defendant's lawyer provide contact information to the clerk and remain nearby in case of a question from the jurors or a verdict but allow lawyer and defendant to leave the courtroom with the understanding that both will not leave the property and travel elsewhere.