Nope, the professor was requesting that you use the standard typesetting software. Some of your classmates might have to write many papers, and hence it's good that they start learning.

Apparently. It's a shame that researchers hae to use such software. Probably part of why research is so conservative and slow. Why not a funky html "paper"? With interactive formulae, clickable links (hey, wasn't PageRank based on references? Maybe time to turn that around) to references. Most important, a clickable link to an academic profile.

Please explain to me how to type the fraction (a^2+b^2-c^2)/(2ab), which is the simple $\frac{a^2+b^2-c^2}{2ab}$ in TeX, so that the numerator appears above the denominator in HTML. Or any non-TeX layout.

A div/span with vertical aligment containing 2 divs, of which one has border-bottom or border-top set.

Please give an example of this. What's the markup? It's almost certainly more long winded and harder to read.

Now make it machine-searchable.

You can put clickable links in LaTeX.

And if you're including mathematical formulae, which scientific academia will do, HTML is rubbish for formulae.