The answers are (1) maybe; and (2) definitely, no.
Texas was admitted to the union by an 1845 joint resolution of Congress. It was a unique admission, and one of the unique aspects is that the joint resolution states that Texas shall have the right to, in effect, subdivide into as many as five different states. If it did so, the most important effect would be to increase its representation in the U.S. Senate from two to as much as ten.
However, this provision of the joint resolution may be unconstitutional. Article IV, Section 3 of the Constitution provides that "no new State shall be formed or erected within the Jurisdiction of any other State...without the consent of the [Legislature] of the [state] concerned as well as of the Congress." Thus, for instance, if, today, the California Legislature decided to divide itself into the two states of Northern and Southern California, it could not do so without the consent of Congress. It seems unlikely that the Senate, in particular, would give such consent because it would have the effect of diluting the power of the sitting senators.
But what if Texas--right now--decided to try the same thing; and instead of one extra state, it tried to create four? Texas has one thing that California does not--that 1845 joint resolution of Congress. If Texas unilaterally purported to divide itself into five states now, without the present consent of Congress, there would be a good argument that this attempt would violate Art. IV, Section 3 of the Constitution, quoted above. If the consent of the current Congress were required, it would seem unlikely that it would be obtained. [The only scenario where I could imagine consent would be if we had a situation like that which existed during most of the Bush/Cheney administration, where both houses were controlled by one party and that party would be the one that would appear to benefit from the new senators.]
The counterargument would be that Congress has already given its consent--it did so in the 1845 joint resolution. There is no way to know which theory would prevail; I suspect that the Supreme Court would decide that to interpret the joint resolution that way would put it in violation of Article IV. But if that is so, then the concession given to Texas in 1845 was illusory. There is a principle of statutory construction that an act of Congress should not be interpreted so as to make it a nullity. But when Constitutionality is at stake, it cannot be bound by rules of statutory construction. If an act of Congress is unconstitutional as interpreted, then it has to be a nullity, irrespective of the intent of the Congress at the time.
So, I think the stronger argument is that such action would require the consent of the contemporary Congress; but it is, at least, debatable.
It might be added that Texas has never shown any great desire to subdivide. As a sixth generation Texan by three different family lines, I think I know why: pride; particularly pride in being large; indeed, until the 1958 Alaska admission, the largest. Everything in Texas is bigger, as they say. Our state song pointed out that we were the largest until the Alaska admission. I remember going to an assembly of my entire elementary school in the third grade to be told the sad news that we would have to change the song.
It is true that in 1969, Texas State Senator Red Berry of San Antonio introduced a bill in the Texas Senate to divide Texas into two states, north and south, with San Antonio to be the capital of the south. There had been several referenda just prior to that time on legalized gambling in Texas. The legalized gambling proposition always lost, but it did have a majority in the southern part of the state. A theoretical South Texas would have legalized gambling of which Senator Berry was a strong proponent. I worked for the Texas Senate at the time and watched as this bill was considered. It died for lack of a "second."
The second question related to a right to secede. It seems that there has developed, in some Texas circles, the misconception that the admittedly unique aspects of the Texas admission documents also conferred the right to secede. In early 2009, in two separate publicly noted instances, Texas Governor Rick Perry implied that Texas had such a right.
Among the people who think that, I wonder how they square that with the events of 1861 to 1865. As we know, in 1861, Texas did, in fact, secede. Well, I suppose an argument might be made that losing on the battlefield does not mean that one's legal position lacked merit. But the fact is that no one at the time even brought up any alleged "right" of Texas, in contrast to the other Confederate states, to secede. No one brought it up because it did not exist.
It also happens that there is a U.S. Supreme Court decision, rendered shortly after the Civil War, which speaks directly to the legality of the Texas secession. It is Texas v. White, 74 U.S. 700 (1868) where the Supreme Court found it necessary to speak to the legality of the purported Texas secession in 1861 because the case involved the rights of some holders of bonds issued by the "Confederate State" of Texas. The majority opinion is by Chief Justice Salmon Chase, and it is the law of the land even today. Its holding that the Texas secession was illegal could not possibly have been stated in any stronger terms:
"When, therefore, Texas became one of the United States, she entered into an indissoluble relation. All the obligations of perpetual union...attached at once to the State...[I]t was the incorporation of a new member into the political body. And it was final...
"...[T]he ordinance of secession, adopted by the convention and ratified by a majority of the citizens of Texas, and all the acts of her legislature intended to give effect to that ordinance, were absolutely null. They were utterly without the operation in law. The obligations of the State, as a member of the Union, and of every citizen of the State, as a citizen of the United States, remained perfect and unimpaired."
In any event, it is clear that Texas has no more right of secession than does any other state--that is, it has none.