You may be too young to remember but, once upon a time, there were these quaint machines called typewriters; you might want to start your research with them, then perhaps move on to the chain-train printer (another relic).

Quite soon after the invention of typewriters, which necessarily had monospacing, type foundries released fonts in imitation.

But were there any monospaced fonts before typewriters?

Like oldnick said, they came about because typewriters and early printers were easier to manufacture if they just spaced the letters out evenly.

Those early devices weren't designed to create elegant layouts with fine-tuned spacing; they were just a way to get words on a page.

In handset type, as well as Linotype and Monotype, numerals intended for use in tabular matter (as in an annual report) were monospaced so that the columns would align regardless of which numeral you put in them. Monospaced numerals like this were referred to as "lining figures," to distinguish them from "old style figures." The latter have ascenders and descenders and variable set widths, and were designed for use within text paragraphs.

Lining vs. Oldstyle has nothing to do with monospace.
Figures were always monowidth, AFAIK, before the switch to lining c.1800.

Which was the first font to have proportionately spaced figures?

I'm not sure that this really counts, but while not all fonts of initials are monospaced, some are, with every letter being placed in a decorated square box. That might be the only instance of a monospaced typeface being created before there were typewriters to imitate.

I do remember from a type specimen book a monospaced typeface that was not created to imitate a typewriter. It was called "mailing list type".

Page 537 of the Pacific Type Specimen book from ATF in 1896 shows it.

In western typography, perhaps the first monospaced fonts were indeed in imitation of typewriters. Certainly I've seen such in early ATF specimen books (circa 1900).

But if one looks to Asian ideographic fonts used for Chinese, Japanese or perhaps Korean, I would expect you'd see monospaced fonts centuries earlier. I don't know if those are interesting for your purpose.

Perhaps it's because I have never used a typewriter, but monospaced fonts always remind me of computer screens of the 80's? Now, I wonder, did the people who made computers had the monospaced fonts in their systems on purpose, because of the screen capabilities or maybe monospaced pixel fonts were easy to make for them and that's why they used those kind of fonts. Or, maybe it was the screens that showed the fonts monospaced and pixel like?

Early computer screens generally used monospaced character displays. It simplified the software and hardware, which was not very fast and had very little memory compared to modern computers. This "character mode" was usually the default. The character shapes were hard-coded into the hardware. Displays were advertised as having, say, an "80-column display", meaning you could fit 80 characters across the screen (usually about 24 lines). The size was usually fixed, although sometimes there were "low res" modes that doubled the height and/or width of the characters.

In some computer systems (maybe most?) you could change out the default character designs to a set that was stored in memory, but this meant you had less memory for programs. Usually, you only did this for games or graphics programs.

Proportional fonts and other sizes were possible if you programmed the computer to use its "graphics mode", which let you draw arbitrary images on the display and address each pixel independently (you could only address characters in character mode). However, this used a lot more memory and was a lot slower than character mode.

There were computers in the '70s and early '80s that had proportional fonts, such as the Xerox Alto and preview displays for computer typesetting front-ends. There were also graphics programs for the IBM PC and Apple II that let you work with proportional fonts for word processing and graphics. The Mac (1984) was the first popular personal computer to use proportional, arbitrarily sized fonts by default (no character mode--graphics mode only!), but it also included a monospaced font (Monaco). (I seem to recall that some reviewers knocked it for not having an "80-column display".)

One thing that was probably also a factor in favor of monospaced fonts in early computers: Before CRT displays were economically feasible, you usually interacted with the computer using a teletype terminal. This was basically an electric typewriter that could transmit and receive character codes to and from the computer. It was natural (and technically easier) to emulate the monospaced typewriter output on the CRT display, rather than emulate typesetting.