Learn English – Why “USSR” but not “UCSR”

Because these usages of shed are assuming a few things about the objects being shed:

a. to cause (blood) to flow by cutting or wounding

b. to pour forth in drops shed tears

Both (a) and (b) require a liquid state in order to flow or drop and (c) is some form of luminance which you noted you don't actually care about. So you could say:

(a) The cyborg shed oil from its veins.

(b) The sky shed rain upon the fields.

These are non-standard in the sense that their usage is extremely uncommon but the meaning still fits.

To directly answer your question: You can shed blood, sweat and tears because they are liquids dispersed from cutting or wounding (blood) or things that pour forth in drops (sweat; tears). If anything else in the human body could do either of those things you could also shed them.

To prove the point:

A urinary tract infection has been plaguing me for days. Yesterday I shed a mere three drops.

By the way, I have no idea where you copied your definitions from but the link you gave doesn't seem to match.

In the journey from Old English to what we write today, the ash (Æ) tended to metamorphose into a simple E and various "ae" forms got reduced to just "e": Ælfwyn became Elvin, Æthelræd became Ethelred, aether and aesthetic became ether and esthetic (except when @Cerb spells them), and so on. The distinction was simply planed off over the centuries. When there was no need for the superfluous Æ (because its sound was rendered with a single letter) it got dialed way down in frequency. The same thing happened, more or less, with the thorn (Þ, þ) and eth (Ð, ð) characters, because the th digraph supplanted them.

An even more obvious influence involved the printing press. In the early days of typography, fonts were imported from Germany and Italy, and those countries did not use the oddball English characters, so substitutes had to be found. "E" substituted quite nicely for the ash, and "Y" for the thorn (as we see on the signs in front of all those cutesy Ye Old Whatever shops).

Elaboration

Asked for citations, I lazily looked to the Web first, but real scholarship in this matter is difficult to Google. Here are some not-stringently-academic citations, together with a disclaimer.

The thorn was particularly popular as a sign for 'th' in Medieval English, but with the advent of printing came a problem. There was no thorn sign in the printing fonts, as they were usually cast outside of England. So, since the sign for thorn slightly resembled the lower-case 'y', that's what was substituted.

The thorn continued to be used, but printing caused its eventual demise from the English alphabet. As mentioned earlier, lingering proof of its existence hangs on in the outmoded 'Ye'.
Thorn — Missing Letter of the Alphabet (reposted with correct glyphs here).

Ultimately, the letter was abandoned when printing began to streamline the alphabet and eliminate unnecessary letters. Æ was separated into AE, and the language moved on. However, you can still find ash used stylistically in names like Encyclopædia Britannica and ÆON.
Mighty Markup.

Disclaimer: I feel it only fair to point out that the reference books I have at hand (printed versions, so no linky-link), suggest that the ash (or æsc in OE), was pretty much gone by 1250 due to the influence of Norman French. This was a couple hundred years before the invention of the printing press, so we cannot accept that as the proximal cause. Still, Gutenberg almost certainly put the nail in the coffin of that and the other oddball characters (including wynn and yogh — look those up for your amusement and edification sometime).