the meanings "mass of small, cryptogamous, herbaceous plants growing together" and "bog, peat-bog" are the same word: Old English meos "moss plant" and mos "bog;" both are from Proto-Germanic *musan (source also of Old High German mios, Danish mos, German Moos), also in part from Old Norse mosi "moss, bog," and Medieval Latin mossa "moss," from the same Germanic source.
These are from PIE *meus- "damp," with derivatives referring to swamps and swamp vegetation (source also of Latin muscus "moss," Lithuanian mūsai "mold, mildew," Old Church Slavonic muchu "moss"). The Germanic languages have the word in both senses, which is natural because moss is the characteristic plant of boggy places. It is impossible to say which sense is original. The proverb that a rolling stone gathers no moss is suggested from 14c.:
Selden Moseþ þe Marbelston þat men ofte treden. ["Piers Plowman," 1362]
Moss-agate "agate stone with moss-like dendrite forms (caused by metallic oxides)" is from 1790. Scott (1805) revived 17c. moss-trooper "freebooter infesting Scottish border marshes" (compare bog-trotter).
"deep mud, bog, marsh, swampland," c. 1300, from a Scandinavian source such as Old Norse myrr "bog, swamp," from Proto-Germanic *miuzja- (source of Old English mos "bog, marsh"), from PIE *meus- "damp" (see moss).
"extreme conservative, one attached to antiquated notions," 1874, American English, used especially of poor rural whites; earlier (1872) in reference to those from the Carolinas who had hid out to avoid service in the Confederate army (and would have stayed out "till the moss grew on their backs"); from moss + back (n.). The same image is behind the use of the word in angling for "a large old bass or other fish" (by 1889).
The dye is obtained from certain lichens. It is naturally blue but turns red in acid and is restored to blue by alkalis. Figurative use of litmus test is first attested 1957, from scientific use of litmus-treated paper as a chemical indicator. Litmus paper with this meaning is from 1803.