Your post doesn't actually ask anything. The implied question is "how shall I look up etymologies and derivations?"
Some of the bigger monolingual dictionaries include more etymological information. I love digging around in derivations, and Shogakukan's 国語大辞典 (KDJ) is a trove of information. I've noticed lately that Kotobank includes some content from this dictionary.
Sites like Gogen Allguide can provide helpful leads. A similar site is yain.jp, although I don't know where they pull their information from, so I take them with a grain of salt. Nihon Jiten provides etymologies for a limited subset of entries, most usefully for the names of plants and animals.
After identifying a likely root, I'll often dig around in the KDJ and other kobun dictionaries to find any likely related terms, and go from there. Sometimes these other entries might specifically reference the term I started from, even if that starting entry didn't mention the later terms. Ah, well...
Separately, you also say, "I believe there should be a better way to arrange these things."
I'd agree. Few Japanese dictionaries include information about etymologies and derivations, and I've had some interesting conversations with native speakers that lead me to think that many Japanese people don't think much about derivations. (That's fair; few speakers of any language really ponder these issues, which is why many of us gravitate to websites like this one. :) )
When I was in my early years of studying Japanese, I was struck by various seeming similarities and I wanted to know more about the etymologies of various words, but I was stymied by the lack of such information, even in reference works. In part to make up for that lack, to make things easier for future learners, and in part simply to scratch my own itch, I've been researching terms and adding such information to entries over at the English Wiktionary. I'm sure there are other similar efforts underway as well.
Side notes:
- 失う is from うし + なう where うし is from ancient verb うす. I think this is more closely related to うしろ (KDJ:「空間的にも時間的にも用い、見えない方をいう」) than to 薄い, since 失う and 後ろ start from the basic idea of "becoming not, becoming invisible", while 薄い is just "thin (but still there)". While 薄くなる is sometimes glossed as "to lose" (such as when said of hair), it really means "to thin out (while still having a presence)". 薄い is probably related to 失う, but I think a bit more distantly.
- 償う is specifically described in the KDJ as arising from older reading つぐのう, from つくのう, from つく + のう. つく derives from betting term ずく, itself from 尽【つ】く. のう is an alternative form of なう.
- 商う is much more probably related to 贖【あが】う・購【あが】う, older reading あかう, あかふ, and 贖【あが】なう・購【あが】なう, older reading あかなう, あかなふ. These are clearly あか + iterative / repetitive ふ, and あか + "doing like so" なう.
The //ak-// root of the あき in あきなう and the あか in あかなう seems elusive. In the older language, we have both あく（飽く・厭く・倦く） "to become full or satisfied", and あく（明く・開く・空く） "to become empty or open", which are almost antonyms. How common are homophonous antonyms in the world's languages? I'm not sure. This has always struck me as pretty weird.
By any gates, these two //ak-// roots don't exhibit any obvious connection to the apparent root //ak-// "to buy, to sell, to barter, to exchange" that we see in あきなう・あかなう. The best I can come up with is a kind of mashing-together of the above roots, "to become empty (by giving something away in exchange) and become full (by getting the other item in the exchange)". That seems ... unlikely, however.
