Can a Supreme Court Justice stay in position judging cases, while
simultaneously being charged with a crime and having to appear in
court themselves for that crime?
Yes.
Indeed, in the U.S. Supreme Court, unlike other courts, a justice is not even required, in any binding manner, to recuse himself or herself from a case in which the justice is involved, although strong judicial norms cause U.S. Supreme Court justices to usually do so when that happens.
There is a norm of conduct that lower federal court judges must take a leave of absence when facing criminal charges, which in the lower federal courts is enforced by having the chief judge of the court in question assign all of the cases of the judge charged with a crime to other judges, that a U.S. Supreme Court justice might actually follow, although that norm would not be enforceable in a binding manner at the U.S. Supreme Court level. At most, the U.S. Supreme Court could decline to assign opinion writing duties and duties regarding ruling on emergency requests to grant or overrule stays of lower court rulings in particular circuits to other justices on a temporary basis. But, the U.S. Supreme Court couldn't deny the Justice charged with a crime a right to vote on petitions for certiorari or on merits decisions of the court.
Is there a mechanism for him to be removed from the Supreme Court
under these circumstances?
U.S. Supreme Court justices cease to hold office upon death, upon their own voluntary resignation, or upon removal from office following a U.S. Senate trial of an impeachment.
But, no U.S. Supreme Court Justice has ever successfully been removed from office via impeachment and no federal official has ever been impeached for conduct that did not occur after the official took federal office.
This said, and despite the fact that impeachment is for "high crimes and misdemeanors" and that the position is held during "good behavior" while in office, an inability to carry out one's duties as a U.S. Supreme Court justice during a period of extended incarceration would almost certainly cause a U.S. Supreme Court justice to resign, or to be impeached if he or she failed to do so. No federal judge has ever continued to serve in that capacity while incarcerated in prison.
If that extraordinary situation arose, verging on a constitutional crisis, because a justice in prison declined to resign, I strongly suspect that the remaining U.S. Supreme Court justices themselves would write a letter to Congress signed by all of them urging Congress to impeach the convicted justice and assuring Congress that their action would be constitutional.
In state courts, judges who are sentenced to prison terms cease to serve because binding judicial discipline systems remove them from office, or because binding attorney discipline systems disbar them and state law or the state constitution requires that judges of the court in which they serve also be attorneys duly admitted to practice.
But, in the Article III federal courts, judges are not required by law or the constitution to be attorneys in good standing (because neither law school nor meaningful regulation of admission to the practice of law existed in the United States in 1789 when the current U.S. Constitution was adopted), and the constitution does not authorize a binding system of judicial discipline that can have the effect of removing a Congressionally appointed Article III judge from office.
Also addressing a follow up question in the comments, there is no mechanism for withdrawing a confirmation of someone whose appointment has been confirmed by the President prior to that person taking the oath of office, other than the death of the person who was confirmed to the post, or the refusal of the person who was confirmed to take the oath of office (i.e. a failure to accept the confirmed appointment).
