If I only want to open and edit LaTeX documents (research papers with equations) and I am using a computer with Windows 10, which of the many versions of LaTeX should I download? Also, is there a way to save LaTeX files so that they are editable in Word or other commonly available programs?
Let's begin at the beginning, shall we?
I ask: What do you mean by a "LaTeX file"? A LaTeX file is written in plain text. It does not show any sort of finished equations. All it contains is coding, some of which might be used for creating equations, tables, and what-not.
So, you edit a LaTeX file in a plain text editor. Don't use Windows Notepad, because Notepad may try to do something that can cause confusion later (it may add hidden characters at the beginning). Don't use MS Word or any other word processor. One simple solution is the free Notepad++ text editor.
But is that what you meant, when you asked the question? Were you thinking of a PDF file, which is the finished output from LaTeX processing? If so, then you are in for a world of hurt. A program such as Adobe Acrobat Pro (not the free Adobe Reader) can, with effort, make some (repeat, only some) edits to a PDF. It is not likely that will be useful to you.
If the edited file will only be printed to paper, rather than circulated as an electronic document, then you can cheat: You can process the PDF pages as images, and alter them in a graphics program, if you know how. No LaTeX involved. Not recommended.
As for needing a LaTeX installation: I believe that ShareLaTeX (online) is likely to suit your needs. But if not, then you can install TeXLive or MikTeX to your own computer. I firmly suggest that you do that as a "portable" installation, to a subdirectory of your user home folder. That way, you do not need any administrative privileges. You are less likely to have a problem with future updates, too. That's because once you get LaTeX working, you can create a ZIP archive of the whole thing, and save it somewhere. Then, if the working LaTeX installation goes wrong, you can revert to the zipped archive.
You will find proponents and opponents for any particular way to install LaTeX, so I don't think there is one best way.
If your editing were only to ordinary text writing, it would be easy to learn what you need in LaTeX, because you wouldn't need much. But alas, if you are editing equations, then you must learn a lot of arcane commands.
EDIT: There is one other possibility. Are you re-writing the material, so that it no longer a research paper, but rather some material from research papers, which will be padded with simple explanatory material for undergraduate use? That is, do you only need to copy some equations into what would otherwise be a rather ordinary word processing document? If so, then there is a relatively easy way to do it:
You can extract the pages from PDF one by one. Then, you can covert each page into an image. The free GIMP will do it, as will some others. The trick is to ensure that the PDF is converted to an image at sufficiently high resolution (pixels per inch, or DPI). A value of 300 is recommended, with anti-aliasing. Use 600 if you have enough computing power.
Then, select an equation, crop, and create a smaller image file containing only that equation. PNG format is recommended. Do this for each equation you need.
Then, you can place the images in ordinary fashion, in an MS Word document. They are not "LaTex" any more; they are mere images, indistinguishable from any photograph.