These are four very similar words, and while native English speakers will likely agree on which one to choose in a given context, they would probably find it difficult to say why. The differences are extremely subtle, and there's a degree of overlap.
Rather than presenting standard dictionary definitions, I found that an etymological analysis helped me get a better sense of the usage:
"in a lower position," early 14c., biloogh, from be- "by, about" + logh, lou, lowe "low" [...] Apparently a variant of earlier a-lowe (influenced by other adverbs in be-; see before), the parallel form to an-high (now on high).
Beneath was the usual word; below was very rare in Middle English and gained currency only in 16c. It is frequent in Shakespeare. As a preposition from 1570s. In nautical use, "off-duty," in contradistinction to "on deck." Meaning "inferior in rank or dignity" is from c. 1600.
According to Fowler, below is the opposite of above and concerns difference of level and suggests comparison of independent things. Under is the opposite of over and is concerned with superposition and subjection and suggests some interrelation.
beneath (adv., prep.)
Old English beneoðan "under, below, in a lower place, further down than," in late Old English "lower in rank, degree, excellence, etc.," from be- "by" + neoðan "below, down, from below," from Proto-Germanic **niþar* "lower, farther down, down" (see nether). Meaning "unworthy of" is attested from 1849 (purists prefer below in this sense). "The be- gave or emphasized the notion of 'where,' excluding that of 'whence' pertaining to the simple niðan" [OED].
under (prep., adv.)
Old English under (prep.) "beneath, among, before, in the presence of, in subjection to, under the rule of, by means of," also, as an adverb, "beneath, below, underneath," expressing position with reference to that which is above, from Proto-Germanic *under- (source also of Old Frisian under, Dutch onder, Old High German untar, German unter, Old Norse undir, Gothic undar), from PIE *ndher- "under" (source also of Sanskrit adhah "below;" Avestan athara- "lower;" Latin infernus "lower," infra "below").
Productive as a prefix in Old English, as in German and Scandinavian (often forming words modeled on Latin ones in sub-). Notion of "inferior in rank, position, etc." was present in Old English. With reference to standards, "less than in age, price, value," etc., late 14c. As an adjective, "lower in position; lower in rank or degree" from 13c. Also
Old English underneoðan, from under + neoðan "below" (see beneath).
This duplication of meanings serves to emphasise an immediacy or directness in the locational relationship. ODO says "Situated directly below (something else)".
So, to very broadly summarise:
- below is more likely to indicate a simple level in an above/below duality
- beneath adds a hint of inferiority or lower order
- under tends to be used in a superposition of related/connected things
- underneath is like "under" but more immediately or directly so.
Regarding your question about "spelling patterns", this is partly explained above in the etymology. To elaborate on "below" and "beneath": they're each composed of the prefix "be-" (a directional indicator) plus a locational word - "low", or "neath" (related to "nether" - think of the Netherlands = "low country").
word-forming element of verbs and nouns from verbs, with a wide range of meaning: "about, around; thoroughly, completely; to make, cause seem; to provide with; at, on, to, for;" from Old English be- "about, around, on all sides" (the unstressed form of bi "by").
Finally, to demonstrate how subtle and indeed fickle the differences are:
- that behaviour was beneath him - but we probably wouldn't use "below" and definitely not "under" or "underneath";
- that performance was below par - or maybe "under par" but definitely not "beneath" or "underneath";
- she slid the key underneath the mat - "under" would work just as well, but "beneath" would be a bit unusual and we wouldn't use "below";
- I'm below the age limit - "under" is probably just as common but seems a bit more colloquial, while neither "beneath" nor "underneath" would be used.