Are you the very model of a modern Major-General?

"I've information vegetable, animal, and mineral."

Amphiboles are a big category of minerals rich in silicon and oxygen that form long crystals. Not to be confused with amphibians, a category of animals including frogs.

"I know the kings of England."

Robert III was king of Scots from 1390 to 1406 and never ruled England. Cnut conquered England and reigned as king from 1016 to 1035. William III of Orange was king of England from 1689 to 1702.

"And I quote the fights historical
From Marathon to Waterloo, in order categorical."

The battle of Salamis was a sea battle fought in Antiquity. The battles of Austerlitz and Borodino were land battles fought by Napoleon in the 19th century.

"I'm very well acquainted, too, with matters mathematical, I understand equations, both the simple…"

2×1 + 3 is 5.

"…and quadratical."

"About binomial theorem I'm teeming with a lot o' news."

I have 8 options for the first topping, and 7 remaining options for the second, but the order doesn't matter, so the total number of combinations is 8×7/2 = 28. The binomial theorem is used for computing numbers of combinations.

"With many cheerful facts about the square of the hypotenuse."

Such as Pythagoras' theorem, which applies to right triangles.

"I'm very good at integral and differential calculus."

Derivatives measures the rate of change. A constant function has the same value everywhere, so its rate of change is 0.

"I know the scientific names of beings animalculous."

Amoebas are soft and extend out arm-like pseudopods. The second image is a radiolarian: it has a mineral shell. The last image is a dinoflagellate: it's like a tiny jellyfish.

"I know our mythic history, King Arthur's and Sir Caradoc's."

The mythical knight Caradoc is also associated with Cornwall.

"I answer hard acrostics."

An acrostic is a word puzzle where squares on a grid are filled with letters, much like a crossword. A sudoku involves numbers instead, and a nonogram (picross) simply dark or light cells.

"I've a pretty taste for paradox."

The other two have non-contradictory resolutions:
Epimenides is a liar, but not all Cretans are.
"This sentence is true." can be either true or false, with no contradiction either way.

"I quote in elegiacs…"

Elegiac couplets are a Greek and Latin poetic form that uses dactyls and some spondees.

"…all the crimes of Heliogabalus."

Elagabalus, or Heliogabalus, was Roman emperor from 218 to 222, and was assassinated at only 18. Elagabalus was high priest of the sun deity Elagabal, and replaced Jupiter with Elagabal as head of the pantheon. This angered Romans, resulting in many accusations, often dubious.

"In conics I can floor peculiarities parabolous."

The eccentricity of a parabola is exactly 1: if you decrease it at all, you get an ellipse; if you increase it, you get a hyperbola.

"I can tell undoubted Raphaels from Gerard Dows and Zoffanies."

Left: Gerard Dow (aka Gerrit Dou), Girl cutting onions (1646).
Middle: Raphael, Portrait of Pope Leo X with his cousins (1518).
Right: Johan Zoffany, Self-portrait as David with the head of Goliath (1756).

"I know the croaking chorus from The Frogs of Aristophanes!"

"Abraxas, galla, galla, tse, tse" is from Jean-Paul Sartre's The Flies (1943). "Ualu ualu ualu! Quaouauh!" is from James Joyce's Finnegans Wake (1939).

"Then I can hum a fugue of which I've heard the music's din afore."

It's fine to transpose music to the range you can sing or hum, but singing in counterpoint to yourself is not possible for most people.

"And whistle all the airs from that infernal nonsense Pinafore."

The other two are from The Pirates of Penzance.

"Then I can write a washing bill in Babylonic cuneiform."

A washing bill or laundry list is a list of items to wash with prices, so it contains a lot of numbers.

"And tell you ev'ry detail of Caractacus's uniform."

He's also carrying weapons, but not much else.

"In fact, when I know what is meant by "mamelon"…"

A breast-shaped hillock, from the French word for "nipple".

"…and "ravelin"."

A ravelin, also called a demi-lune, is a triangular fortification outside of a fortress.

"When I can tell at sight a Mauser rifle from a javelin."

A javelin is a thrown spear, but is also the name of an anti-tank missile. The rifle is a 1871 Mauser rifle, used at the time of the song.

"When such affairs as sorties and surprises I'm more wary at."

Outside a siege, "sortie" generally means any dispatch from a defended position.

"And when I know precisely what is meant by "commissariat"."

A commissariat is a department of the army that provides food and fuel. This is part of logistics.

"When I have learnt what progress has been made in modern gunnery."

Sam Colt patented his revolver in 1836. The Gatling gun was invented in 1861, and the Tommy gun in 1918.

"When I know more of tactics than a novice in a nunnery."

Tactics is about manoeuvres for executing a plan.

"In short, when I've a smattering of elemental strategy."

Strategy is about large-scale decisions and setting broad goals.

"You'll say a better Major-General has never sat a gee."

That is, ridden a horse.