Today, we cover:
The Latin itself is not complicated. E.g. abbreviates exempli gratia, "for the sake of example." I.e. abbreviates id est, "that is." Both entered English writing centuries ago and never left, and both are short enough that nobody forgets how to spell them. What people forget is which one they meant. Merriam-Webster reports that these are the most commonly looked-up abbreviations in English — a statistic worth sitting with, because it says the confusion is not a failure of vocabulary. Readers already know both translations. They look them up anyway.
The reason is that the two abbreviations do opposite things to a set. E.g. reaches into a group, pulls out a representative item, and leaves the group open — the reader understands that more members exist and were not named. I.e. does the reverse. It restates the whole thing in different words and closes it, promising the reader that nothing has been left outside. Write "the citrus fruits, e.g., lemons," and you have offered one instance of many. Write "the citrus fruits, i.e., lemons, limes, oranges, and grapefruit," and you have claimed to have named them all. Choosing between them requires knowing whether you are sampling or defining, and that is a question about your own argument, not about Latin.
Every mechanical rule follows from that. The Chicago Manual of Style keeps the periods in both abbreviations, follows each with a comma, and precedes each with an opening parenthesis, a semicolon, or a dash. In formal prose it confines them to parentheses, notes, and tables; everywhere else, Chicago would rather you spell the thing out in English. The instruction that reveals the underlying logic, though, is the prohibition on writing e.g. and etc. together. If e.g. has already signaled that the list is a sample, then etc. adds nothing — it announces a fact the sentence has already announced. And etc. after i.e. is worse than redundant. It is a contradiction, because i.e. claimed completeness and etc. immediately withdraws the claim.
The conventions surrounding the abbreviations are less settled than the meanings. Bryan Garner's LawProse lesson lays out the split plainly: American usage precedes e.g. and i.e. with a comma or dash, keeps the internal periods, and always follows with a comma. British usage drops the periods entirely and typically drops the trailing comma as well, so that eg sits bare in the sentence. Garner's argument for caring is professional rather than aesthetic — in legal documents, punctuation that obscures the relationship between listed items is a liability, and a redundancy like and etc. reads as carelessness to exactly the audience you were trying to persuade.
What holds across all three sources is the ordering. The distinction between sampling and defining is fixed; the punctuation around it is a house style that changes with the publication and the country. Writers tend to invert that, treating the commas as the hard part and the meaning as obvious, which is how a sentence ends up with the correct punctuation wrapped around the wrong abbreviation.
There is a way around the whole problem, and Chicago already recommends it. Outside of parentheses and footnotes, write "for example" or "that is." The English costs a few characters, carries no risk of transatlantic mispunctuation, and forces you to decide what you meant before you write it.
The abbreviation was never the hard part.