This is pretty irrelevant in graduate school unless you are looking at specific, higher-level graduate courses that are not necessarily found most other places.
This is not irrelevant, and as long as the school you are eyeing is a good research fit, that is the most important consideration here.
This is also completely irrelevant. You are not going to get a liberal arts graduate degree, nor are you going to take liberal arts courses as part of your engineering graduate degree.
Generally speaking, this is true. Ivy League schools have historically not viewed engineering as a a pure enough science to warrant academic study and left it to the land-grant universities to teach it. It has only been in relatively recent decades that they started putting more effort into it, but they don’t have that history that many of the state and polytechnic schools do. Several are good (e.g. Cornell and Princeton) and others have individual departments and research areas that are really good (e.g. Harvard’s biomedical engineering), but otherwise, on average, engineering is just not really a big thing at those schools.
When it comes to a PhD, your employment prospects depend much more on who your advisor was than on what school you attended. It’s the connections made through your lab that will most likely lead to your future job.
I wouldn’t necessarily go that far. Plenty of top programs and top research groups in their respective fields are based at, for example, state schools, which are far broader than STEM-based schools. These are all good options as well.