The State of Texas has no power to stop the Border Patrol from doing anything.
The 10th Amendment allows them to do this.
Isn't immigration the federal government's territory? Doesn't the federal government have the power to act against the will of the states and not the other way around so long as the federal government isn't doing something unconstitutional?
You are right. In the first instance, it is the federal governments responsibility.
I don't know the US Constitution very well, but the 10th amendment refers to a section saying that the states are allowed to take control of their borders when
"the federal government is not doing it's job properly".
I would hazard a guess that 11,000 people per day qualifies as
"the federal government is not doing it's job properly".
Clearly open to interpretation, but I do know this is their argument.