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Most international airlines don’t have first class, they’ll have economy, premium economy and business class.
Virgin Upper Class is business class and it is lie flat.
The way to be sure, and to see for yourself about a product, is when you have found potential tickets, take the airline and plane flight model and YouTube/Google search it.
Not all business/first class is created equal.
Go and see what the experience is actually like on board.
There are many travel creators and flight reviewers who cover this stuff extensively.
And for flights to Paris, keep searching.
If your dates aren’t completely rigid, you should be able to do better than 160 each way.
Try searching Air France without putting in date so you can see their calendar and see where you get.