There's a trivial association between mood and Ger. Mut, in which "bravery" is just a circumstantial entailment of the social expectations there and then entailed by the moral ethos of the time.
Unmut still rather corresponds to mood, in the bigger semantic field of "moral", "attitude", etc. Ger. müde conversely means "tired, sleepy", perhaps not from the same root (cf. *mew-, assume dimm, fuzzy), but certainly seen in connotations of moody (~ having Unmut).
Therefore, the adjective with -y must be secondary, half built from the noun, half from the root in müde, and thus not directly equivalent to mutig.
Further, mood might be comparable to modus, too (cp. rile-midel vis-a-vis ethics). Whereas Mut contrasts with Wut (anger, part. or adj. wütend; Tollwut, adj. tollwütig), Hut (alertness, chiefly in auf der Hut sein, cp. hut, house, v. hüten, behüten *(s)kew- "to cover, protect").
Further appearant compositions that have nothing much to do with bravery are Anmut (adj. anmutig, part. anmutend), Schwermut (adj. schwermütig), Missmut (adj. missmutig), Mutwillen (adv. mutwillig, part. mutwillens), Gemüt (literally the facility that embodies mood; cp. Gemächt, Macht, *might [and magic], …) and gemütlich (comfortable (e.g. of a couch), pleasant, light (of a stroll in the park). Also cp. Vergnügen, genügen, genug, Genüge versus modesty, moderation.
Further, motzen, to mutter, and several more entries in the Lexicon under m- might become conflated, initially, who knows. [TODO: what do Kronnen say?] As far as I know, Middle High German Umlautung (Schwebelaut) is not fully understood. [PS: I had first written
missmütig under influence of schwermütig--that's how easy these can change. I noticed when proof-reading, but still had to confer the spellchecker, lol.]
Conversely, German brav rather exclusively describes an obedient, well mannered subject (of children, dogs, etc). It's probably a loan from Romance languages (viz. bravo), I assume, thus uninteresting. But it shows an analogic semantic development in English.
Further comparisons, e.g. with Beruf (calling, job-title, burden) read as obedience (call and response, responsibility), thus further Dienst (work, service) would be highly speculative in search of a parallel in Mut.
Also always confusing: courage, care and Lat. cura are not appearantly related [wiktionary].