Middle English roum, from Old English rum "space, extent; sufficient space, fit occasion (to do something)," from Proto-Germanic *ruman (source also of Old Norse, Old Saxon, Old High German, Gothic rum, German Raum "space," Dutch ruim "hold of a ship, nave"), nouns formed from Germanic adjective *ruma- "roomy, spacious," from PIE root *reue- (1) "to open; space" (source also of Avestan ravah- "space," Latin rus "open country," Old Irish roi, roe "plain field," Old Church Slavonic ravinu "level," Russian ravnina "a plain").
Old English also had a frequent adjective rum "roomy, wide, long, spacious," also an adverb, rumlice "bigly, corpulently" (Middle English roumli).
The meaning "chamber, cabin" is recorded by early 14c. as a nautical term; applied by mid-15c. to interior division of a building separated by walls or partitions; the Old English word for this was cofa, ancestor of cove. The sense of "persons assembled in a room" is by 1712.
Make room "open a passage, make way" is from mid-15c. Room-service is attested from 1913; room-temperature, comfortable for the occupants of a room, is so called from 1879. Roomth "sufficient space" (1530s, with -th (2)) now is obsolete.
mid-14c., "associate, fellow, comrade;" late 14c.,"habitual companion, friend;" from Middle Low German mate, gemate "one eating at the same table, messmate," from Proto-Germanic *ga-matjon, meaning "(one) having food (*matiz) together (*ga-)." For *matiz, see meat. It is built on the same notion as companion (which is thought to be a loan-translation from Germanic). Cognate with German Maat "mate," Dutch maat "partner, colleague, friend."
Meaning "one of a wedded pair" is attested from 1540s. Used as a form of address by sailors, laborers, etc., at least since mid-15c. Meaning "officer on a merchant vessel" is from late 15c.; his duty is to oversee the execution of the orders of the master or commander.