Paris In August. How Much Will Be Closed?

Not much is closed, certainly not enough that you won't be able to find a boulangerie, cafe or restaurant close to where you are. The only exception might be places that depend on business accounts. The only must see that's closed is Bertillon ice cream parlour

Paris will be full of French (and other European nationalities) tourists.

No museums will be closed, it's peak season for those.
Some boulangeries and restaurants might be closed, but not in a way that would affect a short visit.

We were there last August--all of the tourist sights and museums will be open, but many neighborhood restaurants and shops will be closed. We had plenty of places to eat, but we were staying in a more residential area of the 7th and it was noticeable. Also note that in August some of the trains do not run on the weekends--so getting to/from Versailles for our Sunday reservation was an adventure :)

the August things is mostly myth. Small businesses sometimes take time off then but mostly in outlying districts. None of the tourist sites are closed, most restaurants are open, bakeries don't close (or at least they stage it in a neighborhood so only one at a time closes). Metro/RER run on Holiday schedules on Sat Sun but otherwise should be normal.

Transport and business are a non issue -- what is an issue is heat. I'd never visit this time of year without AC -- we were there in August 2003 when it was 105 for a week running.

Some doctors, dentists, and office workers go on holiday. Some restaurants will be closed but not many. Places that serve tourists will be open. Get an air conditioned hotel room.