According to the Constitution & debates, the legislature of the affected state or the executive (Governor) had to request that assistance. Congress had the obligation to call forth the militia. Lastly, Article IV states that the guarantee of a Republican government only applied to every State 'in this Union' - not those that left.
No it doesn't. The Constituion provides "for calling forth the Militia to execute the Laws of the Union, suppress Insurrections and repel Invasions" by Congress. No mention of a need for the state to request that assistance. Likewise, Section 2 of the Militia Act says "That whenever the laws of the United States shall be opposed or the execution thereof obstructed, in any state, by combinations too powerful to be suppressed by the ordinary course of judicial proceedings, or by the powers vested in the marshals by this act, the same being notified to the President of the United States, by an associate justice or the district judge, it shall be lawful for the President of the United States to call forth the militia of such state to suppress such combinations, and to cause the laws to be duly executed. And if the militia of a state, where such combinations may happen, shall refuse, or be insufficient to suppress the same, it shall be lawful for the President, if the legislature of the United States be not in session, to call forth and employ such numbers of the militia of any other state or states most convenient thereto, as may be necessary, and the use of militia, so to be called forth, may be continued, if necessary, until the expiration of thirty days after the commencement of the ensuing session." So you're wrong in that respect.