1. Research shows women who are cared for by their son's wife in later life fare less well than those looked after by their own daughter
Is "that" not required after "Research shows"? Also usage of 'later 'seems to be a bit awkward
I would not reckon the use of that as mandatory, although use of that will improve the flow. But as far as I see, there is nothing ungrammatical about skipping it. I feel American English bent on brevity doesn’t much bother about using that .
What will be a better word that can replace later life? Older life? I can’t see much that makes an impacting difference here.
2. In fact, the set-up is so unhealthy an elderly woman would be better off living by herself
Usage of "that" after "so unhealthy...”. Is "better off" a right usage. (Although it’s commonly used)
Yes you do require that to complete the idiom properly. Unhealthy is used in a metaphorical sense and not in a literal sense, I suppose. Because better off denotes material wealth and is the comparative degree of well off. I wonder what the connection is between being unhealthy and being better off, unless the sentence tries to invoke the proverbial health is wealth concept. I would rather omit the off.
3. Their analysis revealed that living with a son and his wife to be highly detrimental to a woman's health.
Usage of that(.......to be...). Should we use "is" highly detrimental
Yes, most certainly. Their analysis revealed is the main clause and that living with a son and his wife to be highly detrimental to a woman's health is the subordinate clause. A subordinate clause should have its own verb and to be is not a verb. Hence we do require is as the verb for the dependent clause introduced by that
4. A woman cared for by her daughter-in-law was twice as likely to have died as one who lived with her daughter.
A few observations at first sight.
Is this referring to a particular case or a generalization? This sentence is leaning more on the generalization and needs to be expressed in present tense.
The passive cared for by daughter-in-law is not parallel to the active one who lived with her daughter. In addition the expression likely to have died is weird; likely to die is straighter.