In the old name Dreadnought, is nought an adverb or a noun?
As you surmise, dreadnought originates as a compound of dread and nought, and nought is a noun, so the original meaning was something like shies from nothing. But if you will forgive the use of that most tiresome favorite adverb of pretentious Internet commenters, actually, the first English Dreadnought was a Third Rate galleon commissioned during the reign of Elizabeth I, built in 1573.
Dread is relatively straightforward, meaning to fear, shrink from, be reverent of, and similar senses of having apprehension, from Middle English onwards.
Not (sometimes nocht or noht in older works) is an etymological descendent of nought, but by the time dreadnought appears, it was already a distinct word, and dreadnought is not simply a fanciful spelling of dread-not.
As for naught and nought, the OED draws a distinction between the two as their spellings represented distinct pronunciations until the 17th century or so. They are at least fraternal twins, however:
From the Old English period onwards there is no clear distinction in meaning or function between the two words. (naught adj. is attested considerably earlier than the adjectival uses below, but it is unclear whether this is simply an accident of the historical record, or reflects a real distinction in use.) Usage of one word or the other in any given sense varies relatively freely (and sometimes differs between editions of the same work) down to the 20th cent., although some patterns have been enforced by printers and dictionaries. In early modern English the adjectival senses ‘worthless’, ‘bad’, ‘immoral’ were especially associated with naught (compare naughty adj.), although the use of nought was not uncommon.…
In northern dialects nought generally preserved Middle English -o- ; the vowel of the form nowt… generally has the same sound as that of grow, flown, and other words which had a diphthong /ɔu/ in Middle English.
We no longer use nought in the sense it is used in dreadnought, either. According to the OED, it is a count noun for something which does not exist, a use which is marked obsolete and not attested after a 1642 Henry More ode:
Bringing hid Noughts into existencie, Or sleeping Somethings into wide day-light.
That said, dreadnought began to be used as an adjective in later years meaning without fear:
God is our hope, God is our reuenge, & God is dread naught indeede. (1592, William Burton in Dauids Euidenece)
For this, we have no shortage of synonyms, any number of which the Royal Navy has adopted in turn: fearless, dauntless, intrepid, gallant, undaunted, brave, valiant, and so on. (They never learned what the U.S. Navy learned, that abstract qualities do not have legislators and voters who will vote money to build ships named after them).
I would add that the cloth known as dreadnought was contemporaneously known as fearnought, and in fact the earliest OED entry for fearnought predates dreadnought by more than two decades.
Apart from the use as a name for Martin Guitar, I've never seen it refer to anything but the ship.
But the meaning is clearly fears nothing. As Edwin Ashworth points out, nothing is the direct object of fears.
Etymonline tells us:
Dreadnought
literally "one who or that which fears nothing," from the verbal phrase (drede ich nawiht is attested from c. 1200); see dread (v.) + nought (n.). As a synonym for "battleship" (1916) it is from a specific ship's name. Dreadnought is mentioned as the name of a ship in the Royal Navy as early as c. 1596, but the modern generic sense is from the name of the first of a new class of British battleships, based on the "all big-gun" principle (armed with 10 big guns rather than 4 large guns and a battery of smaller ones), launched Feb. 18, 1906.
Dread
late 12c., "to fear very much, be in shrinking apprehension or expectation of," a shortening of Old English adrædan, contraction of ondrædan "counsel or advise against," also "to dread, fear, be afraid," from ond-, and- "against" (the same first element in answer, from PIE root *ant-) + rædan "to advise" (from PIE root *re- "to reason, count"). Cognate of Old Saxon andradon, Old High German intraten. Related: Dreaded; dreading. As a noun from c. 1200, "great fear or apprehension; cause or object of apprehension." As a past-participle adjective (from the former strong past participle), "dreaded, frightful," c.1400; later "held in awe" (early 15c.).
And yes, as you alluded to, nought is a variant of naught.
Middle English, from Old English nowiht "nothing," variant of nawiht (see naught). Meaning "zero, cipher" is from early 15c. Expression for nought "in vain" is from c. 1200. To come to nought is from early 15c. (become to naught, ycome to naught are from c. 1300).