What is the origin of the basic -다 "dictionary" form for verbs and adjectives? Does it have any meaning on its own? Is it ever used in actual conversations or writing? There are forms that are similar in actual usage such as in the written declarative 해라체 form, exclamations in 해체 form, and reported speech, but those all are conjugated differently depending on if the verb is active or descriptive, so seem unrelated. Did the first dictionary authors just decide to stick a -다 on there? Why not just leave the bare stem (e.g. 먹 instead of 먹다). Of course that gets a bit messy with some irregular verbs, but still leaves me wondering why just a plain 다 was added from a historical perspective.
The citation form of a verb is a particularly specific construct, tied to the requirements of a dictionary user. For inflecting languages such as Korean, it is very important that the user can deduce the citation form of the verb from any conjugated form found "in the wild".
In the case of modern Korean, the citation form is a very simple but slightly artificial form: stem + 다. This 다, if taken in the agglutinative "each root means something" paradigm, refers to the ''declarative'' mood of basic statements. However, I think this form was chosen for its clarity of form more than its meaning. It's easily broken down into the stem and ending, it's short, it categorises irregular verbs easily (compared to e.g. 해체).
One should not confuse the dictionary citation form with the similar-looking 해라체 (often called the "plain" form or the "non-polite formal" speech level), which appear similar and even the same for descriptive verbs (stative verbs / adjectives) but are different for processive (action) verbs, as the original questioner has pointed out.
If we look at the history of Korean lexicography, by 1880 when the Dictionnaire coréen-français was published the concept of the "infinitive" dictionary form had already been established, corresponding to the same stem + 다 form used in more modern times.
Going back to the Gyerim Yusa (雞林類事), a kind of glossary for the Chinese Song dynasty envoys of the speech of Gaeseong from the early 12th century, we have a fairly random set of verb endings used there. The most common though is the form ending in -ra -라, written with 囉 in the glossary, either an imperative or a declarative in Early Middle Korean. So it must have been in the intervening period that it emerged.
The process of lemmatisation for inflecting languages has always had a certain element of arbitrariness, and its only through the process of standardisation and publication that it becomes set as "the one way". English for example has certain verbs which do not have a true infinitive, such as shall; in this case, the present tense (which has the same form for all three persons and both numbers) is used. In Latin and Ancient Greek, although both have infinitive forms, the standard used in dictionaries is the first person present indicative active. In Hebrew, the most common is the third person masculine qal perfect. The principal thing that these forms have in common is that they are among the shorter forms of the verb in their respective languages.
The question why the root is never listed on its own is an interesting one too. I would hunch a guess at a European tradition of requiring the ending (as the ending contains essential conjugation information in e.g. French, Spanish, Italian, without which one could not conjugate correctly). But in the absence of any further historico-lexicographical information I can't be too sure.
Is there any language out there, where in the dictionary(ies) of it, one can find only the root form of the word?
"-다" has specifically long and complicated history with roots going back to the Middle Ages, but one must also understand, in modern Korean language, "-다" is the most basic and simple degree of politeness. Therefore, it was chosen.
I... as one of the non-specialist Korean speakers... think '-다' is chosen for the dictionary form for a simple reason.
'-다' contains no additional meaning to the word!
Compare these examples containing one or more pre-final verb endings each to add a special meaning to the verbs. 놀다 vs 놀리다 놀지않다 놀고싶었다 놀겠다 놀았다 놀던 노는 놂 놀았음
...'-다' form can be found in news headlines. <레알 마드리드, 무패 행진: 바르셀로나를 넘다> <대전 '알뜰주유소가 더 비싸다'> Considering that news headlines often use simple and abstract forms of language... this examples might help you understand that '-다' is thought to be the simplest form for any verbs or adjectives.