As to West Virginia, from this recent General Questions thread :
Technically, it (that is, the federal government) didn’t.
What happened was–the existing government of Virginia proclaimed that the state had unilaterally seceded from the Union (as had several other states at that time), confirmed by a popular referendum. The federal government of the day decided that this (unilateral secession) was unlawful and unconstitutional and was a rebellion or insurrection against the lawful authority of the United States. After the secessionists opened fire on a U.S. military installation (Fort Sumter), Lincoln called for the loyal states to provide troops to supress this rebellion.
Meanwhile, in Virginia–specifically in the very far northern part of the state, in Wheeling , in what is now West Virginia–conventions of Virginia citizens opposed to secession met and declared that the actions of the state government were unconstitutional and illegal and that the offices of the state government were therefore vacant. A new government, the “Restored Government of Virginia” (which I’ll call the RGVA) was therefore proclaimed, to act as the state government for the entire state of Virginia (including the areas of the modern state of Virginia and of the future West Virginia). In practice, the previous state government (which had in the meantime joined the Confederate States of America) continued to exercise actual control over most of the state; Unionist forces never lost control of the areas directly adjacent to Washington, DC, and much of what is now West Virginia remained in Unionist hands–in much of that northwestern area, the popular vote had been against secession–and under the control of the RGVA.
Even before the outbreak of the war, there had been intra-state regional tensions in Virginia and calls for the creation of a separate state in the northwestern areas; the RGVA therefore recognized the results of a popular referendum held in areas which it controlled late in 1861 to create a new state, eventually named West Virginia, which was duly admitted by Congress as a state of the Union with its own state government. The RGVA continued to exist as a rump government loyal to the U.S., ostensibly governing all of Virginia which had not been ceded to the new state of West Virginia, but in reality operating (to a limited extent at least) in areas which remained under Union military control throughout the war (e.g., the city of Alexandria).
There was precedent for a state government consenting to the division of its state and Congress then admitting the new state–before 1820 Maine was part of Massachusetts . That said, there were undoubtedly some questionable aspects of the whole thing. But it was more complicated that the federal government just unilaterally dividing a state into two states, which would indeed be unconstitutional .
As to states being “readmitted to the Union”: The Union government’s position was that the attempted secession of the Southern states was simply an illegal act of insurrection; according to the post-war Supreme Court decision Texas v. White