Meaning in Context – What Does Put Mileage on Somebody Mean?

everybody gave them the road, and took their ribald insolences meekly, without venturing to talk back

Gave them the road is not an expression that native speakers of my part of the U.S. would say, so it seems outdated. As a speaker of 20th-21st century English I would interpret it to mean something like

everybody yielded them the right of way

which means, everybody got out of their way and let them pass by or whatever.

There is idiom in Cassell's Dictionary of Slang give someone the road meaning to avoid, to ignore and another entry for the same ohrase that is an alternate version of give someone the bag. When you look at that one (p. 603), none of the definitions really fit. The date given for the first entry (avoid, ignore) is 1910s+, which is weird, since Twain published P&P about 1880.

I am not sure how well ignore or avoid fits here, since the passengers also took their ribald insolences meekly, without venturing to talk back. Maybe one of the things they did was to avoid or ignore them. This definition seems to fit, and again the phrase give someone the road is not used today in my experience. But avoid can have a meaning of yield someone the right of way.

Whatever it means, it means to try to stay out of their way. Sometimes native speakers have to guess too.

In English, the idiom "wears many hats" means to have many roles or responsibilities. For example,

She wears many hats: she's a doctor, a musician, and a writer.

means that this person has many roles in life (a doctor, a musician, and a writer). The 'hat' is synonymous with 'job' in such expressions.

Therefore, "Putting on my detective's hat" means that the person stepped into the role of detective. That's not to say that the person is a real detective - that depends on the context of the passage - but it means the person started asking questions, as a detective would.