I'm sorry I wont be able to provide any answer other than: "Don't do it!"
Other than using some of the converters that you and other people have suggested, which may give you a reasonable place to start, there is really no satisfactory solution to convert an existing Word document into a LaTeX one. This is difficult to implement, and for a good reason. Word and LaTeX have two completely different approaches at describing and encoding documents. Word (as most people use it) stores formatting information while LaTeX encodes the "logical structure" of a document (which then gets translated to TeX formatting instructions while compiling). Trying to get a LaTeX document out of a Word document is somewhat similar to trying to obtain a C++ program from it's binary: You can do it, but the results wont be pretty.
You are better off taking a printout of your assignments and typing the whole thing yourself in LaTeX. Of course you can copy/paste large bodies of text, but for equations there is nothing better than typing them yourself. If you do this for a while, and show your beautiful documents to your teachers and colleagues, you might be even able to convince them to switch to LaTeX all together.
LaTeX is not a format to store pretty documents, it is a fully-fledged system to create those documents.
Pandoc will convert most major formats to both PDF and old Microsoft Word (doc). LibreOffice can then be used headless.
For the font issue, there is the --reference-docx=FILE option.
Use the specified file as a style reference in producing a docx file. For best results, the reference docx should be a modified version of a docx file produced using pandoc. The contents of the reference docx are ignored, but its stylesheets are used in the new docx.