You think that if a thousand cartel guys showed up at a nuke plant with the intention of melting it down that armor and airstrikes would help matters? I think that would just help them actually.
The Mexican cartels are notably wary of messing with U.S police agents and police/military/border security agencies. There are isolated cases of very foolish sicarios or smaller, more irresponsible organized crime cells playing at threats against U.S border agents or at U.S agents inside Mexico but it's rare and often discouraged brutally by the larger cartels. The simple reason why is they know that with the U.S government, the subversion they regularly practice with pathetically corrupt Mexican authorities works much, much less and goes completely out the window if a U.S police agent dies by their hands. The U.S government strongly underscored this with the cartels by spending decades and God knows how much money diligently pursuing every last person they could find who had any connection to the Kike Camarena murder. The lesson hasn't been forgotten by the bigger cartel bosses, even today.
I absolutely do think they could. They operate vast smuggling networks in the US right now with near impunity, as one proof of this capability. Also, US intelligence is pretty bad when it counts. It didn't catch 9-11 or the fall of Afghanistan or the Soviet Union. Why would it detect a raid on a nuke plant?
...Yes. If you kill the people attacking the sensitive site, they will be unable to attack the sensitive site on account of being dead.
A thousand man force is a battalion. The cartels have no real armor and no air support. No heavy weapons. No training.
The US Army fields 31 brigades. A brigade is 5 battalions.
For every cartel member in your theoretical force, there are ~149 trained, armed, active duty service members in deployable condition for combat roles.
That doesn't include artillery, sustainment, the US Airforce or the US Navy. I am also not including guard teams which are another 27 brigades.
Your theoretical cartel force would also have to go through the Texas State Guard (1,700 people) and US Border Patrol.