Basically the title, under US Federal law were slaves considered citizens? If they weren't citizens under the law, what was their legal status?
Also, (I am not sure if this requires another thread) could slave owners take their slaves with them when they traveled? Did ambassadors bring their slaves with them to their foreign posts?
The court cases of Dred Scott answer two of your questions. Dred Scott was a slave in Alabama, but was bought and moved by his new master to Illinois, a free state, and again to the Wisconsin territory, a free territory. Later, he died and his wife moved to St. Louis (slave territory), leasing Scott out again. Scott then sued for his freedom in Scott v Emerson.
Scott's legal argument was based on the precedence that his and his family's presence and residence in free territories required his emancipation.
So to your question, slave owners could take them with them when they traveled, yes, but if slave were taken to free territories, a slave could sue and win their freedom.
Scott likely would've won the case, but he could not prove he was enslaved by his master's widow. He lost, appealed, and was granted a new trial. He won that trial by finding proof that he had been enslaved by his master's widow. His master's widow then appealed that decision to the Missouri Supreme Court.
The Missouri Supreme Court ruled that Scott and his family were still slaves and should've sued while in a free state.
At this point, Scott sued his current owner at the federal level in Scott v Sandford. There were three questions before the Court:
Did the Supreme Court have jurisdiction to hear Scott's case?
Was the Missouri Compromise of 1820 in compliance with the Constitution or an overreach by Congress?
Did residency in free territories or states entitle a slave to freedom?
The court ruled that "that neither Scott nor any other person of African descent—whether or not emancipated from slavery—could be "citizen of a state", and upheld the Compromise of 1820.
Finally, the court ruled that "upon a careful examination of all the cases decided in the State courts of Missouri referred to, that it is now firmly settled by the decisions of the highest court in the State, that Scott and his family upon their return were not free, but were, by the laws of Missouri, the property of the defendant; and that the Circuit Court of the United States had no jurisdiction, when, by the laws of the State, the plaintiff was a slave, and not a citizen."
Slaves were ruled to be property.
Yes, people did bring slaves with them when they traveled to foreign posts. For example, Jefferson brought Sally Hemings with him to Paris. In practice, slaves brought to France were free, but they'd have to leave their owner, or their owner would take them back to the United States.
Each free state had a different law regarding if slaves brought there were free, and those laws changed over time. For example, New York's law was that slaves were free after 6 months' residence in the state. This was significant because New York was a popular state to vacation in for long periods of time, by wealthy southerners, and they did bring slaves with them. In other states, and at other times in New York State history, slaves were free the moment they arrived in the free state. At other times, they were free as long as they had lawfully arrived in the state (i.e. had not escaped and had been brought there by their owner). These legal differences were very contentious, and were a central issue regarding slave states' assertions of "state's rights". They wanted their own laws to apply in the free states, basically, so the free states 1) couldn't just say that escaped slaves were free, 2) couldn't prevent Southern states from going to other states to take possession of their property, and 3) would use legal forces in the state to assist them in getting back their property.