The ABC order already existed in some form about 1400 BC, in the Ugaritic script, from which our alphabet is descended. From Wikipedia:

It is unknown whether the earliest alphabets had a defined sequence. Some alphabets today, such as the Hanuno'o script, are learned one letter at a time, in no particular order, and are not used for collation where a definite order is required. However, a dozen Ugaritic tablets from the fourteenth century BCE preserve the alphabet in two sequences. One, the ABCDE order later used in Phoenician, has continued with minor changes in Hebrew, Greek, Armenian, Gothic, Cyrillic, and Latin; the other, HMĦLQ, was used in southern Arabia and is preserved today in Ethiopic.[16] Both orders have therefore been stable for at least 3000 years.

The English alphabet comes from the Latin alphabet (it is even often still called the Latin alphabet), which in turn comes from the Greek alphabet. All modern alphabets are most probably in some way descended from the Proto-Sinaitic alphabet, which probably originated around 2100-1800 BC, but whose ordering is unknown. The Greek alphabet developed from Proto-Sinaitic through Phoenician. Many times in between Proto-Sinaitic and English, letters have been added and removed. W, U, and J are among the most recent additions, which did not exist in classical Latin.

The first letters or proto-letters were taken by the Proto-Sinaitic alphabet from non-alphabetic Egyptian hieroglyphs, which were mostly ideograms, small depictions of objects. Note that these hieroglyphs were used in an entirely different way—the Egyptians didn't use them as alphabetical letters—, and so the order in Egyptian doesn't tell us much about the order in alphabets. Perhaps there was a logical sequence in those earliest forms, but that is unknown; it might very well be an arbitrary order. For all practical purposes, the modern order is best considered meaningless, though fixed.

Note that the Arabic script is also descended from Phoenician, and its alphabet uses the same order similar to our own.

Yes, quite a few, actually; how you count them I leave up to you.

The Wikipedia article on Old English Latin alphabet mentioned five extras:

In the year 1011, a writer named Byrhtferð ordered the Old English alphabet for numerological purposes. He listed the 24 letters of the Latin alphabet (including et ligature) first, then 5 additional English letters, starting with the Tironian note ond (⁊), resulting in a list of 29 symbols:

A B C D E F G H I K L M N O P Q R S T V X Y Z & ⁊ Ƿ Þ Ð Æ

Taking those last to first...

æsc, ash (Æ, æ)

As in several modern languages including Icelandic, the letter now known as ash, spelled æsc in Old English, was considered a letter in its own right, not merely a ligature between a and e. For example, consider the OE wrǽððu, for modern wrath; indeed the very first letter of Ænglisc itself was an ash, and the very first word of Beowulf has ash in it:

Hwæt! We Gardena          in geardagum,
þeodcyninga,          þrym gefrunon,
hu ða æþelingas          ellen fremedon.
Oft Scyld Scefing          sceaþena þreatum,

ðæt, eth (Ð, ð)

This letter is still used in Icelandic, where it represents the voiced interdental fricative heard twice in English thither. It disappeared from English around 1300.

þorn, thorn (Þ, þ)

Thorn lasted longer than eth, all the way up until Early Modern English. The title of the posting, "Ye olde...", is actually "þ^e olde". It was used in a lot of scribal abbreviations.

ƿynn, wynn (Ƿ, ƿ)

Another letter of the Old English alphabet which we no longer use is wynn, which did get used in non-runic Latin script. Here are three Unicode code points:

 ƿ  01BF        LATIN LETTER WYNN
        = wen
        * Runic letter borrowed into Latin script
        * replaced by "w" in modern transcriptions of Old English
        * uppercase is 01F7
        x (runic letter wunjo wynn w - 16B9)
 Ƿ  01F7        LATIN CAPITAL LETTER WYNN
        = wen
        * lowercase is 01BF
 ᚹ  16B9        RUNIC LETTER WUNJO WYNN W
        x (latin letter wynn - 01BF)

It was used in Old and some Middle English, but went away under French influence around 1300. Like thorn, it’s too confusable with P. Most transliterations of Old English manuscripts replace wynn with w so that it can be more easily read, but there is some resurgent resistence to this. The Ænglisc Ƿikipǣdia (Old English Wikipedia) uses wynn, as you can see in the second word.

Tironian et (⁊)

This old letter, still used in modern Irish, was from a form of shorthand said to have been invented by Cicero’s scribe Marcus Tullius Tiro. Unicode uses U+204A for it. It is a scribal abbreviation for et, just as & is. Byrhtferð listed both ⁊ and & as separate letters of the alphabet for numerological purposes.

Unicode has this to say about the two:

 &  0026        AMPERSAND
        x (tironian sign et - 204A)
        x (turned ampersand - 214B)
 ⁊  204A        TIRONIAN SIGN ET
        * Irish Gaelic, Old English, ...
        x (ampersand - 0026)

Not that unlike the other letters mentioned by Byrhtferð, Unicode considers these two code points to be punctuation.

yoȝ, yogh (Ȝ, ȝ)

This letter was not in Old English at all, but rather is from Middle English. Unicode 1.0 was confused about the difference between ezh and yogh, a distinction carefully maintained by the Oxford English Dictionary. Here are the current Unicode code points and names for these letters:

 Ȝ  021C        LATIN CAPITAL LETTER YOGH
        x (latin capital letter ezh - 01B7)
 ȝ  021D        LATIN SMALL LETTER YOGH
        * Middle English, Scots
        x (latin small letter ezh - 0292)
        x (latin small letter insular g - 1D79)
 Ʒ  01B7        LATIN CAPITAL LETTER EZH
        * African, Skolt Sami
        * lowercase is 0292
        x (latin capital letter yogh - 021C)
        x (cyrillic capital letter abkhasian dze - 04E0)
 ʒ  0292        LATIN SMALL LETTER EZH
        = dram
        * voiced postalveolar fricative
        * mistakenly named yogh in Unicode 1.0
        * uppercase is 01B7
        * Skolt Sami
        x (latin small letter yogh - 021D)
        x (cyrillic small letter abkhasian dze - 04E1)
        x (ounce sign - 2125)
        x (alchemical symbol for half dram - 1F772)

In its article on ezh, Wikipedia says:

In Unicode 1.0, the character was unified with the unrelated character yogh (Ȝ ȝ), which was not correctly added to Unicode until Unicode 3.0. Historically, ezh is derived from Latin z, while yogh is derived from Latin g. The characters do look very similar, and do not appear alongside each other in any alphabet. To better differentiate between the two, the Oxford University Press and the Early English Text Society extend the uppermost tip of the 'yogh' into a little curvature upward.

The definitive treatise on the yogh–ezh confusion is this one by Michael Everson entitled “On the derivation of YOGH and EZH”.

Insular Letters

These are alternate ways of writing letters we no longer write in this fashion. The insular G is particularly distinctive, and when looking for a good uncial script/font, you should pay especial notice to whether it has been reproduced authentically. The insular S looked very much like an R, and the insular T was a C with an overbar.

 Ꝺ  A779    LATIN CAPITAL LETTER INSULAR D
 ꝺ  A77A    LATIN SMALL LETTER INSULAR D
 Ꝼ  A77B    LATIN CAPITAL LETTER INSULAR F
 ꝼ  A77C    LATIN SMALL LETTER INSULAR F
 ᵹ  1D79    LATIN SMALL LETTER INSULAR G
 Ᵹ  A77D    LATIN CAPITAL LETTER INSULAR G
 Ꝿ  A77E    LATIN CAPITAL LETTER TURNED INSULAR G
 ꝿ  A77F    LATIN SMALL LETTER TURNED INSULAR G
 Ꞃ  A782    LATIN CAPITAL LETTER INSULAR R
 ꞃ  A783    LATIN SMALL LETTER INSULAR R
 Ꞅ  A784    LATIN CAPITAL LETTER INSULAR S
 ꞅ  A785    LATIN SMALL LETTER INSULAR S
 Ꞇ  A786    LATIN CAPITAL LETTER INSULAR T
 ꞇ  A787    LATIN SMALL LETTER INSULAR T

Unlike other font variations, these letters are considered so distinct that Unicode assigns them their own distinctive code points. They do not casefold to their non-insular versions, nor do they normalize to such. However, the Unicode Collation Algorithm does sort them close to the more familiar forms. This is the same way it treats ð; therefore, d, ð, and ꝺ are distinct letters that all sort after C and before E in Unicode.

Insular forms are still standard when writing Irish or Scottish Gaelic, although no longer so in English.