There's now a five-time winner in the men's competition

Yes, the red sentence is ok.

Except I wonder if it ought to be five times. Very possibly it ought.

Expressions of this type use the singular of whatever is being counted.

A five-time winner
A two-time loser
A five-dollar cigar
A fifty-dollar watch
A million-dollar prize
A sixty-mile journey
A two-hundred-foot drop

Thank you~

For

In my schoolboy's mind this disparity would refract into an invitation to lie.

Is it better to change SCHOOLBOY'S to SCHOOLBOYS'?

No. The speaker states that he possesses the mind of a schoolboy (singular).

Oh...

Thank you~

Or possibly he is describing an event that took place when he was a schoolboy, and therefore had the mind of one. You neglected to provide any context.

Is it natural and idiomatic to say?:

There's now a five-time winner in men's competition.

Nowadays do native speakers say this?:

-We made a five hours' journey in great heat from Kyoto to Yokohama.

Must HOURS' be changed to HOUR?

It would sound more modern.

The apostrophe part might give the apostrophe freaks a **** hemorrhage. I'll leave that to them.

Oops, no 's', problem solved.

Not this one; a possessive apostrophe in a temporal expression is perfectly acceptable, but often omitted. It goes before the s for single units of time and after for multiple units of time. Some sources advise against the apostrophe after multiple units.

one year's insurance, a week's holiday, a day's work, an hour's delay, a minute's warning

three years' insurance, a five hours' journey, twenty minutes' delay, ten years' wait, two months' salary, six months' interest-free credit

I'd say the first two are American idiom (journalese?). In Britain you probably wouldn't see them written that way; I don't know why exactly.
The other two seem more natural in AmE or real English.

I forgot about a five-pound note, or a "fi'-pun' note" as Dickens had someone say. Or a two-shilling piece from my youth. Or Two-Ton Tessie O'Shea likewise.

Google exact phrase - UK only region
"a five-time winner"
About 238,000 results

"a two-time loser"
About 33,000 results

Try to google 'furnitures' and see what we'll get. I don't know why my laptop can't show the number of results.

"Furniture" is already a plural, so "furnitures" is a bit awkward. But not impossible.

In modern English furniture is an uncountable noun like (for example) advice, accommodation, baggage, bread, cheese, equipment, furniture, information, knowledge, money, pasta, work, progress, research, travel.

We say furniture is expensive or ugly or wooden or absent. Before the end of the nineteenth century, the plural furnitures existed in Standard English in both the U.S. and the U.K.; during the twentieth century, however, it ceased to be used by native speakers. A single item of furniture, such as a chair or a table, is often called a piece of furniture.

A quick Google search for "furnitures" showed me results mostly from non-native speaker sources,e.g. shops in Kenya, Korea, the Phillipines, the Netherlands, India, etc.

Tam o' Shanter, imperfectly remembered

"...it gars me greet
Tae think how many counsels sweet
How many xxx , sage advices
The husband from the wife despises"

Okay uncountable noun then. Bacon.

I'd also like to have google as a convenient tool to learn English, but very often it proves unreliable.

Fashion is an uncountable noun, but often seen in the plural.