"I am expected to start my first semester in college next spring as a math major. While preparing for undergraduate studies I heard from my peers that I should study this language/program called LaTeX which is basically the MS Word of natural sciences."
Your friends are correct that it would be a great idea for you to learn LaTeX if you want to pursue mathematics as your major.
"I went through the math department's curriculum on the website and found zero instances of LaTeX mentioned anywhere."
That means nothing. The department's website is not going to tell you every, single, thing about your upcoming experience.
"No courses on LaTeX, no LaTeX prerequisites, no requirements that assignments must be done in LaTeX."
This is not a wise way to determine whether or not your friends were giving you sound advice. The website also says nothing about you needing to know how to check emails, and doesn't mention any courses on how to check emails, but you will still need to know how to check emails if you want the best chance of performing well during your undergraduate studies.
In almost every university, at least one instructor will offer bonus marks for typing assignment solutions in LaTeX, and for the classes in which no bonus marks are given, LaTeX could be required or at the very least you'll get much higher grades anyway because it's so much easier to read your work when it's in LaTeX versus when it is not. I never required my MATH 135 students at University of Waterloo to type in LaTeX and I never offered bonus marks for it, but the two students that earned the highest grades in the end, just happened to be students that typed their assignment solutions in LaTeX (I wasn't even grading the assignments since the course coordinator decided that they would be graded by TAs, but their assignments were always the easiest to read). When I was teaching biophysical chemistry at Oxford University, I did require my students to type their assignment solutions (though I did not require them to use LaTeX). For upper-year courses in which research-like papers, review essays, end-of-term projects, or undergraduate thesis documents need to be prepared, I would not be surprised if LaTeX is a requirement.
"I emailed some professors to ask whether I need to learn this before I begin my studies a few days ago but no response yet."
By now, hopefully you can see that it is rarely a requirement, but it is indeed helpful for you to learn it and use it. I would not expect an answer from those professors, and if they reply, they will likely say that it's not required, but that does not mean that it would be wise not to learn it or use it as soon as possible.
"Will I be taught how to harness LaTeX in class? Do I need to invest my free time to learn it myself?"
You will not likely be taught how to use it in class, and yes you need to invest your free time to learn it yourself. I worry about your attitude towards education. Are you pursuing this degree because you have a passion for the subject, or because you "need" to do it?
"Can students be penalized for formatting their work in something other than LaTeX?"
Yes they can. Instructors can require you to use LaTeX and can penalize you for not using it. It is unlikely that they will do that, but I recommend that you learn LaTeX anyway and stop trying to get away with just doing the bare minimum that your instructors force you to do.
"I am worried because I have zero prior programming experience."
Typing math in LaTeX is not "programming". You will be typing characters just like you typed them when you asked your question here. I recommend using LyX which is almost exactly like Microsoft Word, except you can enter math equations using the TeX language by pressing CTRL+M to create a math box. You will find typing math much faster and easier once you start using TeX, for example whenever you want to type the symbol for the Greek letter "pi", you'll just have to type
\pi in a math box.