Question: > 
> They inflicted gunboat diplomacy on these poor island people.
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> 
> 

Inflict means "to cause something harmful against", so this doesn't seem the right word.

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> They forced gunboat diplomacy on these poor island people.
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> 

This doesn't seem to be correct. A more neutral way to say this would be:

> 
> They practiced gunboat diplomacy on these poor island people.
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> 

What's the correct verb to use here? I am not sure forced x on them is the most appropriate.

Answer: 
"Out of curiosity" is a general phrase. It means you aren't asking the question for a special reason. You're simply curious. It's useless in your example -- the sentence would be better without it -- but it sounds natural enough.

A better example than "a lot of money" is asking a salesperson "out of curiosity, how much does that cost?". You're telling them you don't want to buy it -- you simply wonder about the price. Or "out of curiosity, what does your ferret eat?" means you aren't thinking about buying a ferret yourself. It's common in front of questions where asking them would make the person think you wanted to do something, but you don't. "How long do volunteers work, just out of curiosity?" means it sounds as if you might volunteer, but you don't want to. You're just curious.

If an accountant was calculating your taxes and asked "how much did your company make, out of curiosity?", that would be fine. It means they don't need to know the answer for their job.

The English stack exchange has an [out-of-curiosity](https://english.stackexchange.com/questions/371258/what-is-the-meaning-of-out-of-curiosity) question (but I don't think the examples are as good).
It's a little bit odd, though the meaning is clear enough.

More likely someone would say "Just out of curiosity how much is 'a lot of money'?", possibly shortened to "Just out of curiosity, how much is 'a lot'?" or even just "How much is 'a lot'?"