As it has been suggested that it would be useful to identify all of the Allusions in the quoted piece, I've edited this answer to include further information. And made some discoveries along the way.
Regular, the romin - insulting the kottage injins
The start of this extract begins with a reference to Marcus Atilius Regulus, the roman who defeated the Carthaginians in a naval battle at Cape Ecnomus during the first Punic War.
The OP has already, in comments, identified
Hoikillitz tossin the big bed bull
with the seventh labour of Hercules, wrestling the Cretan Bull
Etlitz jugglin' a lot of woils
Would be Atlas juggling a lot of worlds. Atlas was a Titan condemned to hold up the heavens, but is also credited by some sources with the creation of the first celestial sphere. A celestial sphere in this context maps the constellations on the outside of a sphere, resulting in a mirror image of the constellations as seen from Earth. This is the object that Atlas is typically depicted holding, not a globe of the earth. Hence he is juggling the many worlds of the celestial bodies.
Ajex knoggin' the lightnin' around
This appears to refer to
Ajax of Oileus (after his father, Oileus), or Ajax the Lesser, because he was not the equal of the Telamonian Ajax In the sack of Troy, [Ajax] violated Cassandra at the altar of Athena, and Athena caused him to be shipwrecked on the way home. Poseidon saved him, but Ajax, boasting of his own power, defied the lightning to strike him down and was instantly struck by it. Other versions of the story say that he stole the Palladium and that later Poseidon destroyed him for blasphemy.
"Ajax." The Columbia Encyclopedia, 6th ed.. . Encyclopedia.com. (July 19, 2019).
It actually seems that most versions of the tale I can find don’t talk about him being killed by lightning. The phrase ‘Ajax defied the lightning’ seems to have has great currency at one time, but I’ve not identified a source, though it was used by PG Wodehouse in Summer Lightning. The earliest use of the phrase I can find is in a book from 1845 ‘Will Watch: The Bold Smuggler: A Tale of the Coast : the Narrative Founded on Fact, and Characters Drawn from Life. Where it is linked to Ajax, son of Telamon rather than Ajax, son of Oileus.
When Ajax, the son of Telamon, whom Homer sometimes likeneth unto a ass, and you, good reader, have, without a doubt, seen worthily represented by the merry Andrew Ducrow, in his Hippodramatic exhibition at the foot of Westminster bridge, on the Surrey shore of the River Thames – we say, when Ajax defied the lightning, he would never have been such a fool had he known anything about he principles of electricity; for he would have then known that the lightning, properly treated, was a harmless well-conducted body enough.’
Onto the allusions that the OP specifically inquired about:
may be a reference to Horatius Cocles who
defended the Pons Sublicius from the invading army of Etruscan King Lars Porsena of Clusium in the late 6th century BC, during the war between Rome and Clusium.
as immortalised in the narrative poem 'Horatius at the Bridge' (link kindly provided by @user14111, I had remembered enough of the poem to add 'Tiber' to my search terms but hadn't though to link the poem itself)
Some sources also note that the nickname 'Cocles' indicates a one-eyed man, or cyclops.
Cyklops destroyin' the tutty-nine dumb dimmins
I’d guess that ‘dimmins’ are demons, this is supported by a paper by Dr Taimi Olsen in 2008 where they state
Krazy, acting like Jack Dempsey, falls into the role of a street-smart guy and uses boxing slang (“K.O.” for knock-out) and casual Northern speech (‘regla” for “regular” and “golla” for “golly”)—and an immigrant reference (“knock it for a goulash”). He (she is not very feminine in this cartoon!) brags that he is a battling demon or “bettling dimmin.”
I remain in the dark as to the significance of there being 39 of them, or any particular tale of Roman Mythology that relates demons to cyclopes.
I suspect that it may be related to Polyphemus; Odysseus fools him with the aid of strong wine, hence Pupp's response 'Skol', a toast. However that still doesn't get us to 39 demons.
In Roman mythology the cyclopes forge Jupiter's thunderbolts.
Jupita darin' the tunda bolds
Jupiter was obviously the God of Thunder, so the association seems clear. ‘Daring the thunderbolts’ I would read as his having (per the OED) the ‘boldness or courage’ [to hurl] his thunderbolts.
However, looking at the letter substitutions employed in Krazy Kat's speech, it may be that we aren't looking at 'daring' at all. Looking within the same sentence, we have the word 'bolds' for 'bolts'. Substituting 't' for 'd' in 'darin' gives us 'tarin' which could, in Krazy Kat's ideolect represent 'tearing', which is a recognised description of one of the sounds that thunder makes.
Energy from a lightning channel heats the air to around 18,000 degrees Fahrenheit. This causes the air to rapidly expand, creating a sound wave known as thunder. The stepped leader causes the initial tearing sound
A snapping or tearing sound before the main thunder is caused by a failed leader, a streamer of positive charge going up from the ground.
So we could be expected to understand 'Jupiter tearing the thunderbolts', but I'm not 100% convinced by this argument myself.
Nobil - movillis - killosis
taking some hints from this book Comics and Language: Reimagining Critical Discourse on the Form which invites us into a
Mental recitation of these often mystifying spellings in order to hear what actual words stand behind the wretched phonics of “movillis -stipenditz-killotzil”...’
which I eventually decoded as ‘marvellous -stupendous -colossal”, suggesting that the phrase you asked about would be ‘noble - marvellous-colossus’.
So. That quote back in the Ajax section, which rumbled on with a dozen of more commas and didn't seem to add a great deal? It turns out that that was the key to unlocking this whole list of allusions and why Krazy Kat is calling them out as they do.
Merry old Andrew Ducrow performed an act called ‘The Living Statue or Model of Antiques’ during which he assumed
‘various poses including several that imitated Hercules battling the Nemean Lion’(Wyke 1997b:53-54)
This knowledge leads us to a body-builder named Eugene Sandow
Sandow’s act did not consist of any marvels of strength. Instead, covered in plaster dust, he struck a number of poses in imitation of classical statuary. A number of famous works such as the “Dying gaul” were evoked, but the most famous imitation, and the one that became his trademark, was his imitation of the Farnese Hercules. His identification with the hero became so strong that in May 1894 he even reprised Hercules’ first labour and wrestled a lion in San Francisco, admittedly a lion muzzled and wearing leather mittens (Blanshard 2005:151-156).’
Sandow became a huge sensation
Photographs of him striking his various poses circulated widely. He even appeared in primitive film. In 1894, he visited Thomas Edison in his studio in New jersey and a short kinetoscope film of Sandow flexing was produced for distribution in ‘Kinetoscopic Parlors’ throughout the country. This was the first time that Hercules became a mass media phenomenon.
Sandow inspired numerous imitators and it was not long before there were dozens of young Hercules snapping at the heels of Sandow and wanting to steal his lion-skin mantle. This was a world wide phenomenon. Originally from Germany, Sandow had devoted followers in Great Britain, France, Canada, Australia and the United States. The fashion for displays of muscular poses in imitation of statuary proved to be so popular that in September 1901 in the Royal Albert Hall the world’s first body-building competition was staged.’
Shows of such 'living statuary' became hugely popular:
The popularity of “antiquity” in the theatrical, literary and educational worlds of the nineteenth century extended even to exhibitions of strength and horsemanship. The strongmen who performed in the nineteenth century were referred to as “Kings of Force” and were often provided with stage names such as Hercules, Romulus, Remus or Cyclops. They carried the classicizing attributes of animal skins and a club or were set in some form of classical scenario such as a “Roman” spectacle, replete with lions, elephants and gladiators. Equestrian programs incorporated circus artists with well developed physiques who rode around the ring in skintight fleshings and posed on horseback in attitudes drawn from classical or pseudoclassical statuary. Criers gave each pose a title, such as “Ajax Defying the Lightining” or “The Fighting and Dying Gladiator”
Therefore I propose the theory that Krazy Kat is not just making educated classical allusions as he identifies these figures from antiquity, he is playing the role of ‘Crier’ as though the ongoing fight is a series of strongman poses as put on by performers such as Ducrow and Sandow.
Given that this takes place in Officer Pupp's dream, and if I've understood what I've read about the strip correctly, it presumably indicates Pupp's desire to be seen as a perfect masculine specimen by Krazy Kat.
As a final thought, Officer Pupp's responses to Krazy Kats identification of poses do seem to be related to the classical allusions, though I can't quite pin them all down. Quince have been identified as being the Golden Apples that Hercules had to steal in his eleventh Labour, Skol is a toast which may tie to a drunken Polyphemus and cave cannam, beware of the dog may be a reference to Atlas' daughter, Maera, was the nymph of the Dog Star, Shis Kebbab highlights the skewering of the challengers from the Etruscan army by Horatius and his two sidekicks...