Graduate school includes the programs after undergraduate college: Master’s degrees, PHDs, JDs, etc. To do cutting edge research, you’ll need a PhD, and any PhD program worth doing will be funded. It means you’ll basically be paid to attend (ok, not well paid at all, but you won’t have to pay anything.)
Check out “where science PHDs got their start”. It’s a list with college names ranked in order of producing people with PHDs in various fields.
At most top universities/LACs, the upper level courses are at ‘graduate’ level because most students enter with AP/IB/AICE/Dual enrollment classes user their belt. It’s their default start, so to speak. The courses are very intense and move quickly, so by senior year the students are very advanced. They then apply to graduate school. The NESCAC undergraduate=> Ivy league path is well-trodden.
At normally selective 4-year universities with master’s or PhD programs, students with AP/IB/AICE/DE classes are considered more advanced than average so they get advanced credit. They often reach senior level classes as juniors (keeping in mind that those may well be junior level classes at more selective colleges). So, senior year, they take classes from the 1st year of master’s classes.
To give you a specific example:
At top colleges, almost all students arrive with calculus classes under their belt, some have taken multivariate calculus. At Penn, the assumption is that everybody who needs ath for their major has achieved a 4 in Calculus BC or at least a 5 in Calculus AB. Students with a 3-4 in Calculus AB are placed in a sort of remedial Calculus class. At a good flagship, a student with a 3-4 in AP calculus AB will start in Calculus 1 or get credit for it - they’re considered on time to slightly ahead, not ‘late’. Finally, at a directional college where most stem students start with precalculus, the same student who d be behind at Penn would be a full year ahead in math.
As a result, the 4th year classes offered to students who start in Calculus 2 or 3, Calculus 1, or precalculus, aren’t going to be the same, and some advanced students who’d be taking the ‘regular’ senior classes at a highly selective LAC would thus be taking the same classes under the title of graduate courses at your state Flagship for example. The big difference is that you may have more choices…or fewer choices… depending on where you attend, but it’s not crucial since you can only take 4-5 courses a semester and even the smallest LAC will have 500 to choose from each semester.
‘liberal’ in Liberal arts has nothing to do with politics. It’s related to the Latin word ‘liberare/to liberate’, to emancipate through knowledge. It’s actually very traditional: it comes from Europe and and the middle ages. At the time, as today at all respected universities, there were seven academic fields nobody could do without if they wanted to be considered educated. These included (translated into contemporary English): English, Speech/public speaking, math, music, critical thinking, Philosophy, and science. All American colleges are based on this original ideal, although they’ve added new branches of knowledge to this core, since many subjects didn’t exist in the middle ages.
As for the way conservatives are treated, it really depends on what you believe ‘conservatism’ to be.
To give you real-life College examples:
Students who feel righteous posting flyers encouraging fellow students to ‘denounce’ (to the police) classmates with Hispanic surnames on account of their being potentially undocumented will be met with anger by other students.
Students who insult, treat poorly, or insist on wanting to ‘cure’ LGBTQ students will be met with hostility.
So, it’s true that if these are your beliefs it won’t be easy at any college.
Students who want to argue for a small government, tax cuts, or fewer regulations on businesses will encounter spirited debates. This is actually par for the course at most colleges 
Students who are libertarians and want to abolish most current prohibitions will also have plenty to discuss with others.
Students who want to defend their religion’s place, or lack thereof, in public life will be treated with respect. (There are places of worship on most campuses and groups for students of every faith to keep growing in that faith, as well as groups for interfaith dialogue.)
Colleges do rescind, often for academic or integrity reasons: the student got D’s or an F, misrepresented themselves in their application, cheated or committed a crime between the application deadline and deposit, it’s absolutely not uncommon.
What you’ve been told about involves a conservative indeed; but the student was recently rescinded, not for being conservative, but because they were supposed to display some characteristics to get in and had successfully hidden some academic/character flaws: do you honestly think that if a student in your US history class used the n-word repeatedly it’d be without consequence? (it was as part of an AP us history study group so I imagine the teacher wasn’t asked for a letter of recommendation); then after all the mass shootings, imagine that same student claimed he wanted to kill a specific group of people who had just been the victim of a hate crime/mass shooting, do you think any college would take the risk?
I doubt you have either claimed to want to kill all x people nor have you used inappropriate language in class (I would hope!) so you would not be at risk of being rescinded.