Firstly, I will note that it's not quite correct that the 时量补语 must be placed before the "last verb" of the sentence. Rather, in those scenarios, the 时量补语 is placed before the last iteration of the verb it applies to in the clause. For instance:
(This year I only went back to my hometown once, because I was too busy.)
Moreover, I don't think it's always necessary to repeat the verb twice.1 For example:
(To prepare for studying at a French university, I studied French for three years.）
In fact, in some cases (such as that of my first example), it sounds more natural for the verb to only appear once. However, if you indeed repeat the verb, the 时量补语 must indeed be placed before the second appearance, not the first one.
That done, I will move onto the specific sentences you asked about:
Notice that the 要 is part of 要学一年: this indicates that it's necessary to learn the subject in question for a year. Thus, this sentence does not mean "she is going to learn Chinese for a year", since this English sentence doesn't convey obligation; rather, absent any other context, it means something like "she will have to learn Chinese for a year."
This means "I only watched the television for ten minutes."2 This works because 只 (only) is attached to 看了十分钟.
Judging from dictionary examples of usage of 能 (e.g. 我能照顾好自己), it is fine to use 能 alone. This phrasing sounds odd to me---I would instead write something like 她能学一年汉语---but I am inclined to assume that this is merely a result of my own preferences.
I am a native Cantonese speaker and have a very intuitive understanding of Chinese grammar in general. It's possible that I am missing something, or that my vocabulary isn't completely standard. I have tried to double-check the key uncertainties I had.
I am not sure what sense of "just" you are using. If you mean that I finished watching the television for ten minutes just now, your translation isn't correct. If you are using "just" in the sense of "only", your translation is fine.