I looked around but did not find that anyone has asked this before, but what are the fonts that are standard/recommended while writing academic reports/papers?
If there's no template, then the choice is yours. However, you should make sure to pick a font that's easy to read. The usual standards in academia tend to be the Times, Helvetica/Arial, and Computer Modern families. This doesn't restrict you from using fonts like Book Antiqua, Myriad Pro, Goudy Old Style, or Garamond, but they're definitely not standard.
For an academic paper each publisher journal have their standards. These do not affect or are affected by the manuscripts sent in to the journal. Some journals specify fonts, commonly standard Times Roman, for their manuscripts. If the journal specifies something, follow that specification. Otherwise use a font that is easy to read. There is no need to use anything but a standard font for whatever typesetting/word processor system.
As others have mentioned, the standard font varies, but is usually a serif font such as Times New Roman, although sans serif fonts such as Arial and Helvetica seem to be gaining traction as well. Their is major disagreement over which is easier to read--serif or sans serif fonts, with no clear consensus on the outcome. For example, see this paper.
Font size is typically twelve point. Follow the guidelines on this one, and make sure to keep your font consistent. Nothing is more likely to get you minus points than some obvious monkeying with the font size, whether to lengthen your manuscript (most commonly seen in undergrad papers) or to fit your text into the page limit (the rest of us!).
There isn't any.
Focus on the content, write using your favorite writing software's default font, and let the journal's typesetting staff worry about the looks of the published version.
For the subset of journals that do not take care of typesetting, first make sure they are legitimate, then use the template they provide.
If no template is provided discuss with your supervisor and colleagues whether the journal is really worth your time, if it is then use your favorite software's default font.