Everyone expects me to turn up as the object of a preposition or a verb […] But me also turns up in a number of place where traditional grammarians and commentators prescribe I. […]
While traditional opinion prescribes someone and I for subject use — I and someone seems a bit impolite — in actual practice we also find me and someone and someone and me […]
Both are speech forms, often associated with the speech of children, and likely to be unfavorably noticed in the speech and writing of adults except when used facetiously.
The Cambridge Grammar of the English Language gives the following four examples to illustrate "Coordinate accusatives corresponding to non-coordinate nominatives", p. 462, commenting that "The accusatives in  are non-standard and strongly stigmatized, especially the pattern in [ii]":
|||i||a.||Tina and me sat by the window looking down on all the twinkling lights.|
|b.||She and us are going to be good friends|
|ii||a.||Me and Larry are going to the movies.|
|b.||Him and me fixed up the wagon while the others went to town.|
But reader SN has pointed out to me that there are some reliable facts about the relative frequency of (some of) the alternatives that these authorities don't directly mention.
Specifically, "me and someone" is about three times more common than "someone and me"; whereas "I and someone" is more than "slightly impolite" — it hardly ever occurs. Thus (with corpus.byu.edu search strings, in which '.' means "start of sentence" and [np*] means "proper noun"):
|. me and [np*]||124||45||3|
|. I and [np*]||2||0||0|
|. [np*] and me||51||16||0|
|. [np*] and I||1395||303||51|
Or here, where "[n*]" means "noun":
|. me and the [n*]||38||3||2|
|. I and the [n*]||8||4||2|
|. the [n*] and me||6||0||0|
|. the [n*] and I||169||28||19|
The pattern for coordinate subjects with he|him ~ she|her is somewhat different. "Him and someone" is again much (five times) more common than "Someone and him"; but now "He and someone" is much commoner (11 times) than "Someone and he":
|. him and [np*]||13||7|
|. he and [np*]||400||143|
|. [np*] and him||4||0|
|. [np*] and he||34||16|
(I've limited this table to the masculine pronouns, since her can also be possessive, and we would need to eliminate common things like "Susan and her mother" in order to include the feminine forms.)
One theory would be that in some varieties of English me is an emphatic form rather than (or in addition to) an accusative form. Thus in the same way that French has "Jean et moi" rather than "*Jean et je", this kind of English has "John and me" rather than "*John and I".
But this in itself doesn't explain why "me and John" is preferred to "John and me" by a factor of three; nor (especially) why the prescription against the order "I and John" (justified on post-hoc grounds of politeness) is so much more strongly obeyed (875 to 1) than the prescription against me anywhere in coordinate subjects (7 to 1).
Obviously, the choice among these alternatives will have very different proportions in different registers, as well as in different regions and classes of speakers and writers. Any serious investigation would need to break these factors down in a systematic way. But I feel that the numbers shown so far are enough to establish that there's something interesting going on here.
One direction towards a solution might be to adopt the theory put forward in Joseph Emonds, "Grammatically deviant prestige constructions", 1986, and adopted e.g. by Nicholas Sobin, "Agreement, Default Rules, and Grammatical Viruses", Linguistic Inquiry 29(2) 1997. This theory says that "nominative" pronouns in coordinate subjects are actually ungrammatical in English ("grammatically deviant") and must be introduced by extra-grammatical editing rules ("grammatical viruses", in Sobin's terminology).
A version of this approach can explain why "I and NP" is hardly ever found, as follows:
1) The natural state of all pronouns in coordinate subjects is accusative.
2) There is a rule (of prestigious deviance) turning "and me" into "and I".
On this view, there's no way to generate "I and someone" (or perhaps there's only another and much more marginal prestigious-deviance rule) and so it should never (or hardly ever) occur. This also explains object forms like "between you and I".
But this theory doesn't explain (without further stipulation) why "he and someone" is so common.
A somewhat different approach is taken by Philipp Angermeyer and John Singler, "The case for politeness: Pronoun variation in co-ordinate NPs in object position in English", Language Variation and Change 15, 2003, which makes a three-way distinction (in object position) among "vernacular" me and X, "standard" X and me, and "polite" X and I.
Also relevant is this back-and-forth:
Frank Parker, Kathryn Riley, and Charles Meyer, "Case Assignment and the Ordering of Constituents in Coordinate Constructions", American Speech 63(3) 1988.
Graham Shorrocks, "Case Assignment in Simple and Coordinate Constructions in Present-Day English", American Speech 67(4) 1992.
Frank Parker, Kathryn Riley, and Charles Meyer, "Here us go again", American Speech 69(4) 1994.
Parker et al. also explain nominative pronouns in conjoined subjects as the result of prescriptivist editing, but argue that the order of conjuncts is explained by general principles of "empathy".
I haven't seen any attempts to survey and explain the extensive corpus data that is now available. This would include looking at the large available collections of transcribed conversational speech, at dialog in out-of-copyright novels now available in digital form, and so on.