This lack of respect for the language of origin not a phenomenon unique to English. When a word is borrowed into one language from another, unexpected things can happen.
I would argue that, for many examples you've given in your question, the actual perception of a singular-plural relationship is messy in practice, and the application of the plural is inconsistent.
Data: Using data as a collective noun with singular agreement is more common than using it with plural agreement. More in another thread from this site.
Alumni: I have heard as many people also use alumni for the singular, or even alum, as I have heard use alumnus for the singular. I imagine my experience with this word is typical (at least in the US), though certainly not universal. In any case, it is messy.
Media: The words media and medium don't even seem to correspond in any meaningful way in actual English usage. The word media has forked off and become a different word entirely. The word media is clearly used as a collective singular noun, as shown in newer constructions like multimedia (not multimedium even though we don't say multistages, multicores, multicycles, multistories, etc.). You will find few people who will ever say "Mass Medium". We talk about someone having "media savvy" even though we wouldn't say "computers savvy" (even though they can work with more than one computer). This is because, in English, these sorts of constructions always use the singular noun, whether it is collective or not. The way that media is used is evidence of how the word is actually parsed, perceived, and used by English speakers.
Another example of how foreign language morphology often doesn't mesh well: people try to pluralize octopus and virus as octopi and viri/virii, respectively. Virus was a mass noun in Latin, where we got the word. The word octopus comes from Greek and would take the plural form octopodes in Greek.
My main point is this: there is only a weak, inconsistent application of this -us to -a or -us to -i to begin with. So forums (like statuses and others) is a word even though we also sometimes have this other rule. Our language seems to continually push us towards either dropping the foreign pluralization in some way or another, or reanalyzing the plural as another distinct word. So I see this confusion as the language trying to mash these words around to make them fit our language naturally.
If we hadn't become so darn literate and knowledgeable in the past few centuries, I imagine these plurals would have regularized by now :)
There are two related problems here. First, the idiom one of the X always has a plural noun as X, even when there is an intervening phrase or modifier, and even if that modifier seems to want a singular head, as is the case with your One of the most common, and generally worst problem[s]. So the simple answer is that problems is correct in both of the sentences given above.
The related problem that you alluded to is the fact that the singular verb is occurs in close proximity to the plural noun problems. This is misleading: the number of the verb is determined by the number of the verb's subject, and the subject of the verb in your examples is One. The fact that one is modified by a prepositional phrase with a plural object does not change the number of the subject.
(There are some exceptions to the rule given above, but none of those exceptions apply to the phrase one of.)
Best Answer
From the comments, it appears that you want to refer to 'some people' for which the pronoun is obviously their) and 'their phones', but use a singular battery. Put like that, it is clear that you will have to rephrase, since there is no correct pronoun for both singular battery and plural phones. Your first example is common, but you should be aware that 'singular they', though commonly used, is also commonly criticised. (e.g. here and here).
Your second is grammatical, but if there is a difference between 'the batteries belonging to the phones' and 'the batteries belonging to the people' (do you really think so?), then it is ambiguous, and cannot be rescued without changing the sentence: when the battery is dead or ...to describe phones with dead batteries would be common rephrasings.
Your third is simply bad English (mixing singular and plural), unless, unusually, all the phones (or, of course, all the people) share one battery.