Paris: Fri-Mon or Mon-Wed

The weekdays anywhere you go museum wise are less crowded than the weekend when a lot of people are not at work.

Crowding at museums is certainly less of an issue when they're closed. ;)

I dunno. You don't clearly state if you are planning to actually go to the Orsay or Versailles. If you are, then going to Paris when they are closed would significantly complicate your planning on a short visit. OTOH if you're not going to those, and just worried that all the people who otherwise would be jammed in those attractions will be out on streets and competing for space with you...I think weekends are almost always more crowded generally.

There are separate questions about sequencing....does Paris make more sense at the beginning or end of a trip? Personally, I like to end a trip on a high note (but, conversely, I often like to slow down a bit at the end of a long trip and savor the last fews days, as I may be a bit worn out from a grueling marathon....so depends on what you want to do when you're in Paris).

No easy answers.

I’d also ask if this will be over the first weekend of a month when many museums are free on the Sunday and extremely crowded.

I would say that in general visiting Paris at the end is more time effective because
- you usually need to spend the last night in Paris if you're flying out of Paris.
- it is a good idea to "push on" to your final destination on the first day, as you land in Europe in the morning and are typically too exhausted for any meaningful sightseeing.

Parisian here, I would say the weekend if you want to visit museums, but definitely not if you want to explore Paris, walk along the Seine etc. The centre is full of tourists at weekends.