Irish (and Newfoundland) English is a bit of a special case -- it's a language that uses primarily English words, but the grammatical rules are at least heavily influenced by Gaelic. You'll see that mostly in colloquial usage (not slang, but the everyday language of ordinary people). You can see that in a lot of places. "I have just done that," for instance, is more likely to be rendered as, "I'm after doing that." SImilarly, the "s" marker tends not to indicate the third person singular, present tense (Standard English), but the
primary verb in a sentence: "I doesn't like it when he do that."
"Methinks" (always one word, by the way) is archaic, and was probably the last holdover from Gaelic in what was to become Standard English (the courtly London-area dialect) other than "do support". Common constructions -- the things we use a lot -- hold on longer than they ought to as the world around them changes. It's the same reason why we still have plurals like men, women and children (words we use often, along with a lot of Germanic farming plurals) when the simple "s/es" plural marker took over everywhere else. ("Do support"? We use the word "do" as a part of the construction of verbs in a way that no other language
except Gaelic does. There are only two other very small regional dialects, one of Italian and one Caucasian, that use a similar construction, and then only in the interrogative.)
That leaves only the case of the compound subject. "Billy and me went to the store." We're all taught in school that it's wrong, but there is good evidence to suggest that it is, in fact, correct. "Me went to the store" has always been wrong in Standard English, and the two (and there were only two of them) pedantic old cranks who almost simultaneously decided to apply Latinate grammar rules to English (to make it a respectable, world-class language) in the seventeenth century decided that the compound needed to change as well. Where did they go wrong?
The compound "Billy and me" is an expansion of "we". "We", in a self-respecting Indo-European language, has always expanded to the object form. That is, the answer to "who are we" has always been "we are you, him and me". These two idiots decided that it needed to be "we are you, he and I". In most Indo-European languages, you would say "it's me", not "it is I". No French speaker would ever say "c'est je"; the correct form is "c'est moi". But then French already had the juice; they didn't need to put on respectable airs for company.
Unfortunately, three hundred years of blackboard grammar has most of us convinced that "Billy and me went to the store" is wrong, and to the majority of educated people, it does sound wrong -- even if it is the form that best fits our internal grammar (the ordering of branching in the
semantic trees is remarkably consistent in any natural language that isn't in the process of changing word orders, and English is a "hard" SVO language). So, in day-to-day conversation, don't worry about it, but in written communication, it's probably best to go with the flow, so to speak, just in order to maintain the veneer of academic respectability.
But outside of Ireland, Newfoundland and (perhaps) parts of Australia, one ought not use an isolated "me" as the subject of a sentence.
"/me" is a variable -- in the elder world of chats, BBSs and dungeons, it was replaced with your handle (user name). At first, seeing it naked online meant the user had forgotten where he/she was at the time, then it became a badge of leetness. Now it's a joke of sorts, used to describe what your imaginary avatar would be doing at this point in an online conversation.
/me takes a sip of coffee, sparks another smoke and laughs quietly to himself at the thought.
Note that all of the verbs in that statement are in the third person. That's correct usage for the "/me" construct.