I agree with Imperator that rural areas are referred to as "country" (e.g., "My mother retired to the country.") In this context, the opposite of "country" would be "city" (just like the opposite of "rural" is "urban"). However, "town" is also used to refer to suburbs, which are neither urban nor rural.
Within a rural or suburban area, "town" can refer to the business district. In the popular book Little House on the Prairie by Laura Ingalls Wilder, Laura and her sisters go to school "in town" (where the stores and lumber mill are) from their homestead farm. This usage is consistent with your last three sentences:
In town particular care is needed because cyclists and pedestrians may
not hear it coming.
In town, when you're walking along a road, you don't make eye contact.
In town is another story, as its massive bulk can make it unwieldy in
tight parking areas.
Your first sentence seems odd to me (a native speaker of American English):
In town, most women do domestic chores and child care while their
husbands are at work.
To answer your question, I would not consider "in town" to be the opposite of "in the country". To me, at least, "in town" might be the opposite of "on the outskirts" (which would include the country and sparsely populated residential areas). The expression "downtown" can also be used in place of "In town" in your last three sentences.