The word former refers to state. It means that the person or object was something, but no longer is. The word "former" only refers to sequence in special cases where it is contrasted with the word "latter" (in which it means the first of two given items).
The word previous refers to sequence. It means that the person or object was/did something before something else took over or replaced it. As StoneyB rightly clarifies in his comment, "the previous" means the directly preceding member in a series, however "a previous" can refer to any earlier member in the series.
The word last, in this context, means the same as "previous", but is only used to refer to the immediately preceding item. (Jay warns that "last" can be ambiguous as it is also used to refer to the final entry in a series.)
Thus, you can say "former coworker" of someone who is no longer a coworker. However, the word "previous" means the one before the current, so saying "previous coworker" or "last coworker" does not make sense unless you had a coworker and they were replaced.
Referring to TV series, you can say that you preferred the "previous" or "last" show, and this would refer to the show immediately preceding the current show. You cannot say the "former" show in this context.
When referring to a show that aired earlier than the immediate previous, you can say "a previous show". Otherwise, you can be more specific. Depending on the circumstances, you might say "the first show", or "one of the earlier shows", or "episode ten", or simply "one of the older shows". You could even say something like "five episodes earlier".
In this context, manner and way are identical in meaning. Dictionary references are not particularly helpful, as the relevant meaning of each is defined in terms of the other. For example, the Oxford dictionary defines way as A method, style, or manner of doing something. Manner is perhaps slightly more formal.
They live in a beneficial way through education.
Whichever is used, the resulting sentence is flawed and meaningless. What the writer probably wants to say is:
Their lives are better because they are educated.
The problem is that beneficial is an adjective that describes the source of the benefit, not the recipient. In this sentence, education is the source of the benefit. We can move education to the front, and then through is not required. The recipient (their lives) needs to be attached using a preposition for or to. The sentence can therefore be rewritten as:
Education is beneficial to/for their lives.
The adverbal construct in a xxx way is much more widely used in Arabic (bi-il-taryqah xxx) than in English, as natural adverbs are few and far between in Arabic: English written by arabic speakers therefore tends to contain this construct even when it's not necessary in English. If the writer really wanted to stick to an adverbal form, then the English adverb well would be a perfect choice:
They live well through education.
They live well because of their education.
Best Answer
The sentences are not different. According to most dictionaries they have more or less the same definition when used as a plural noun.
They both refer to older people, but the primary difference is perception and interpretation. Generally speaking it sounds more eloquent to use 'elderly' than to use 'old'.
vs.
Using 'elderly' is a more polite, and sensitive way to address an aging population.
As far as how old someone must be to fit into that category, it's all subjective and relative so there is no definition for that. Whatever you consider old, is old.