It all depends on the university - different ones operate in different ways.
Berkeley does admission by college, and, in the College of Engineering, by major. For example, they state that it is harder to get admitted to electrical engineering and computer science than it is to get admitted to mechanical engineering, even within the College of Engineering. [url=<a href=“http://admissions.berkeley.edu/selectsstudents%5DHere%5B/url”>http://admissions.berkeley.edu/selectsstudents]Here[/url]
UCLA does admission by college and major within the School of Engineering and Applied Science. You have to apply to a specific major. You can also apply “undeclared,” but apparently admission to the undeclared major is also very competitive. UCLA doesn’t say but I would imagine that their electrical engineering and computer science majors are more competitive than the other engineering majors. [url=<a href=“http://www.admission.ucla.edu/prospect/Adm_fr/fradms.htm%5DHere%5B/url”>http://www.admission.ucla.edu/prospect/Adm_fr/fradms.htm]Here[/url]
For UCLA and UC Berkeley, it’s not 100% clear whether if you are rejected from your major of choice you are also rejected from the university as a whole. But, if you are rejected from CS as a freshman, I would say that even if you could still attend transferring into the CS major is probably nearly impossible.
UCSD has this unique college system that’s akin to a residential college; there are 6 general colleges to which every UCSD student is sorted. The college determines your general education requirements; the engineering school determines your major requirements. So admission is a two-step process: First, you apply to UCSD, and specify that you are applying as an engineering major. If you are admitted and assigned to a college, then the engineering school evaluates your application. Some engineering majors are open; if you are admitted to UCSD you are automatically admitted to that major. Some engineering majors are impacted; you have to be evaluated for admission to those majors, and you can be denied. So at UCSD it’s possible to be admitted to the university but not to the major. Of course computer science and electrical engineering are both impacted majors at UCSD. [url=<a href=“http://www.jacobsschool.ucsd.edu/admissions/admissions_freshman/steps.shtml%5DHere%5B/url”>http://www.jacobsschool.ucsd.edu/admissions/admissions_freshman/steps.shtml]Here[/url]
So at UCSD it is very possible to be admitted to the university, but not to an impacted engineering major. However, if you were admitted, you could still major in an open engineering major. However, right now it looks like ALL of the UCSD engineering majors are impacted. They’re all listed on [url=<a href=“https://students.ucsd.edu/academics/advising/majors-minors/impacted-majors.html%5Dthis%5B/url”>https://students.ucsd.edu/academics/advising/majors-minors/impacted-majors.html]this[/url] website. So in practice, this becomes a case in which you could get admitted to UCSD, but not any engineering majors, and then find it very difficult to transfer into an engineering major later.
At Washington there are three different ways to be admitted into engineering. You can be admitted directly from high school into an engineering department by specifying an engineering major as your first choice on the application form. UW says that only 10-20% of engineering majors are admitted directly into a department. If you are not admitted into a department, you are admitted as a pre-engineering major. Secondly, you can apply for “early admission” to an engineering department at the end of you freshman year, if you have strong grades. Thirdly - the most common way - you can apply for admission to the engineering departments in the middle of your sophomore year, after completing the pre-engineering prerequisites.
At UW it is also possible to be admitted to the university, but not engineering. Furthermore, UW seems like one of the few places on your list where not being admitted to engineering as a freshman is not a bad sign, but the norm, and that you’d still have two more chances to transfer into engineering at UW.
Michigan and UIUC appear to do admissions similarly - you have to specify that you are applying to the College of Engineering even though you apply to the same undergraduate admissions office as everyone else. They don’t specify that certain majors are more competitive than others, though. UT-Austin also has you apply through the regular admissions office, but you do have to specify a first- and second-choice major, and it’s implied that admissions outcomes may vary depending on what major you specify - some are more competitive than others.
At UT-Austin it sounds like it’s possible to be admitted to the university but rejected from engineering. It’s not clear whether that’s the case at Michigan and UIUC. It’s also not clear on how easy it would be to transfer from the colleges of liberal arts and sciences to an engineering major, if you get admitted to the university but rejected from engineering. I would imagine given that engineering is so competitive at these universities that it would be difficult.