Is there any etymological relation between "a-hunting" and "ajar"?

While reading this question I recalled hearing the phrase like to go a-hunting on several occasions when someone stylized their speech to sound old-time'y and now I started wondering if there is any relation to other words, like:

• afloat, ajar, apart,
• abroad, aback,
• afoot, aboard, afield,
• aback, afore, afront,
  etc.

While I have little or no idea about their individual etymologies and overall evolution of the language, I perceive this class of words as description of state formed by "a"+attribute where the attribute is more-or-less related to the state (pretty direct relation in afore or afloat, but no so obvious in others like ajar or along, if we even consider the latter to fit in this class as well).

Then we have to go a-hunting. Hunting itself may here be perceived as a noun describing an action, and in that form it would form sort of state description: the person went and is-hunting, or as form of desire/necessity: the person went in-order-to-hunt, with the latter being related to the state-of-being-hunting as well.

My questions are:

• were afloat, abroad, etc written (or spoken) a-broad, a-float at some point of time?
• at the time when to go a-hunting was used, was something like to be a-hunting also used?
• is it just a coincidence, or is there any evolutionary relation between forms like afloat and a-hunting?

According to Merriam-Webster's Eleventh Collegiate Dictionary (2003), words with an a- prefix fall into two major categories: those in which the prefix reflects the Greek alpha privative for negation, as in abiogenesis, achromatic, and ahistorical; and those derived from Old English a in the sense of "on, in, or at," as in abed, aloud, afire, and atingle. The words abroad, afloat, a-hunting, and ajar fall into this second category of a-prefixed English words, and so are related.

The Eleventh Collegiate further distinguishes four cases of the Old English–derived a- prefix:

a- prefix {ME fr. OE} 1 : on : in : at {abed} 2 : in (such) a state or condition {afire} 3 : in (such) a manner {aloud} 4 : in the act or process of {gone a-hunting} {atingle}

The Eleventh Collegiate indicates that afloat and ajar come from the first subcategory of a- words (the ones where a- signifies "on, in, or at"), noting the following etymologies for them:

afloat adj or adv {ME aflot, fr. OE on flot, fr. on + flot, fr. flot deep water, sea; ...}

and

ajar adj or adv (earlier on char, fr. ME, fr. on + char turn — ...)

It doesn't cover the origins of abroad, but that word seems to follow the same pattern. MW cites a-hunting as an example of the fourth subcategory of words, where a- means "in the process of."

Michael Quinion, Ologies and Isms: Word Beginnings and Endings (2002) offers some additional detail on the sense of a- in words from non-Greek roots:

a- Towards, of, in, into, or at; marking some ongoing process or state; movement onwards or away. {Old English prepositions of or on (sometimes as unstressed an), or the Old English prefix a-.}

The Old English prepositions were originally separate words, but became reduced to a- and attached to the words they once modified. The process can be seen in alive, which in Old English was two words, on līfe, literally 'in life'; others of similar type are aside, akin, and anew. Some examples are verbs derived from Old English a-, which had an idea about it of an action or an intensification of an action: arise, abide, and awake.

Some objectives imply a continuing or active state, and have much the same force as a present participle ending in -ing [cross reference omitted]: ablaze, abuzz, afire, afoot, aglow, astride. Others combine the prefix with a present participle, usually hyphenated; such words imply an ongoing processor activity: a-brewing, a-roving, a-hunting, a-wasting. Though they are mostly now archaic, literary, or dialectal, the form has had a small revival in recent decades, as in Bob Dylan's song lyric The times they are a-changing.

As for the other queries asked near the bottom of the posted question:

• It appears that afloat, ajar, and perhaps abroad, were at some point in Middle English rendered as two words: on flot, on char, and perhaps on brood. In a series of library database searches, I found 27 instances of a-float (between 1621 and 1698), 1 instance of a-jar (from 1694), and 10 instances of a-broad (between 1611 and 1698). So these words did pass through an intermediate period during which some writers fused the prefix into the root word, but marked the fusion with a hyphen.

• At least in the case of a-hunting, the earliest matches that I could find for the term were from the 1600s—and 15 of the 16 from that century were from 1650 or later. The exception is from William Slatyer, The History of Great Britanie (1621, page 213): "But deafe as dumbe, and wanton as/ Light lyther aire, more faine she was/ To seeme more faire, right woman too,/ Spreads all her Peacocks plumes to woo,/ Fresh as the morne fond Nymph, to gaine/ Light loue! her spouse a-hunting's slaine." (That is, her husband is slain while a-hunting.) But an example in Thomas Bromhall, An History of Apparitions, Oracles, Prophecies, and Predictions ... (1658, page 112) includes the phrase "he seemed to be a-hunting to those that beheld him often times"—which provides an affirmative answer to the poster's question "at the time when to go a-hunting was used, was something like to be a-hunting also used?" An example of "to go a-hunting" appears in Adam Olearis, The Voyages and Travells of the Ambassadors Sent by Frederick, Duke of Holstein, to the Great Duke of Muscovy and the King of Persia (1669).

• I noted at the end of the first paragraph of my answer that all of the non-Greek a- forms discussed in this answer are related through the Old English prepositions that both Merriam-Webster and Quinion mention as sources of the a- prefix.