Yes, a sitting president can be sued for legal matters, but there are limitations and legal procedures that must be followed.
No, you cannot sue yourself for any legal matters as a person cannot be both the plaintiff and the defendant in a lawsuit.
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The official that advises other officials on legal matters is the Attorney General. He or she is responsible for ensuring that no arm of the government breaks the law in any way.
Not in the US or any country that I know of. Dueling is illegal and it matters not who challenged whom.
No. The US Constitution specificaly states that the sitting president's term of office ends at noon on 20th January. The president-elect may take the oath of office anywhere he happens to be just before noon, and be sworn in by almost anyone serving in some sort of official capacity. No emergency can delay that process.
Adams was a lawyer by profession but I do not think he did any legal work after he was President.
Dwight Eisenhower
Sitting means "current" or "active" president. According to the smarties on StackExchange politics, it is acceptable, to call ex-presidents "Mister [lastname]", or "Former President", or even "Mister President" - even though they no longer have that role. Famously, Marilyn Monroe addressed a sitting member of the executive staff (President Kennedy) as "Mister President" in a happy birthday song, and most guys who listened to that.. well, their members were standing rather than sitting afterwards. ;)
No, it is not allowed for any living president to be on US bills or coins. You may see some companies issue a "commemorative" coin to honor a current president, but such coins are for collectors and are not legal tender.
Julius Rosenberg was on death row in 1952. He was not nominated for President by any legal party.
No president legalized homosexuality. States gradually changed their laws against homosexual practices sometimes due to federal court decisions.
No, a sitting president cannot be arrested for criminal offenses they may have committed while in office. They can only be impeached by Congress and removed from office.