Spending 4 Nights/3 Days in Paris

Some thoughts:

The Paris Museum Pass is either two or four days, so doesn't fit three days very well. Your second day is very busy and you'll only have time do get a flavour of the Louvre. It's also pointless to visit Versailles, imo, unless you're spending a full day there.

Champs-Elysees is a dull, traffic-clogged street full of chain stores and overpriced cafes. Definitely see the Arc de Triomphe, but don't bother walking down it, get a bus if you really want to see, or just take the metro.

The Orsay is also very large and you may not have time or energy to fit in another museum. The Rodin Museum is currently under reconstruction and only a portion of the collection is on show.

My personal suggestion is to get a two-day Museum Pass, which may not save much money but will give you the important option of visiting the Louvre on both days, or twice at the beginning and end of days, so you can explore bits of it in depth and leave when you get bored. Drop Versailles if you are only in Paris for three days. On the morning of the final day, if you are interested in Impressionist art, visit the Marmottan Museum, which is a private museum that is not included in the PMP, but has a really good collection.

Try to squeeze in Champs Elysees & Arc de Triomphe after you arrive from London (you will have plenty of daylight).
Tuesday, do your free stuff or places not included on the museum pass.
Then you would only need a 2 day Paris Museum Pass for Wednesday+Thursday.
Take advantage of the late closing times - Louvre on Wed and Orsay on Thu.
Unfortunately you will need to rush Versailles - get an early start.
http://en.parismuseumpass.com/
Of course you know to buy a 'carnet' of ten cheaper priced tickets to use on the Metro.
The Bat-o-Bus is a good way to get between places along the Seine (and much cheaper than the other boats).
http://www.batobus.com/en.html

Monday 15 June
Eurostar to Paris, Champs Elysees, Arc de Triomphe

Tuesday 16 June
0900 Eiffel Tower
1200 Notre Dame
1500 Sacre Coeur & Montmartre
1800 Seine River Cruise

Wednesday 17 June
0900 Orangerie (Monet's waterlillies) 9am-6pm. Closed on Tuesdays.
1100 St Chapelle (amazing stained glass) 9:30am-6pm (late opening until 9:30pm on Wednesday).
1300 Conciergerie (where Marie Antoinette imprisoned) 9:30am-6pm.
1500 Pompidou Centre (modern art collection) 11am-9pm. Closed on Tuesdays.
1800 Louvre 9am-5:45pm (late opening until 9:45pm on Wednesdays and Fridays). Closed on Tuesdays.

Thursday 18 June
0900 Versailles 9am-6:30pm. Closed on Mondays.
1400 Invalides (Napoleons tomb) 10am-6pm (late opening until 9 pm on Tuesdays).
1600 Musee Rodin 10am-5:45pm (late opening on Wednesdays until 8:45 pm). Closed on Mondays.
1800 Musee d'Orsay 9:30am-6pm (late opening on Thursdays until 9:45pm). Closed on Mondays.

I think your budget is way too tight. Have you priced hostels and passes?

I was going to say, if you can find a decent hotel in central Paris for 70 euro/night, please let us know.

I've been hunting for airbnb's in Paris in the Eiffel Tower area (for late Sept) and have found many within my budget of $75-100 Canadian (75 can = about 55 euro, but the US has a much better exchange rate)...most are actually whole apartments...I'm sure you could find some private rooms for a bit cheaper - and def cheaper further away from the ET (- if you don't want to do a hostel, tho many airbnb are already getting really booked up, so pickings would be slimmer.)..and yeah, the Champs E is really mostly full of crap stores selling crap stuff to tourists...when in Montmartre - make sure to go to Sacre Coeur (but watch out for the 'friendship bracelet' scammers - they seem to congregate there).

David, a heads up, if you wish to go up the Towers of Notre Dame then you will need your Museum Pass or purchase a separate ticket for that. The Towers open at 10:00, folks start lining up along the Left side of the Church around 09:00. Once you enter the side entrance you will go to the Gift Shop and from here you can purchase your ticket to the Towers. Then head up for an upper middle view and then to the top for a 360 degree view. The church itself is free, its usually best to enter at 08:30 after the morning Mass, or better yet attend the Mass then tour it.
Enjoy your trip.

I would drop Versailles unless this is a really important thing for you with so little time -- if you do go then go very early so you are in line half an hour before it opens and well before the tour buses arrive.

For the Louvre -- your ticket if you get a ticket and not a museum pass will allow multiple entries on the same day, so if you move that to Wednesday, you could go in the morning and then again late in the day -- it is open till late that evening.

The Orsay (absolutely buy a ticket ahead for this or have the museum pass so you can use the security line for ticket holders MUCH shorter, sometimes no wait at all) is open late on Thursday so that might be a good time to go -- but not if you have spent the rest of the day marching through the crowds and the huge estate at Versailles.

Conde Nast featured the Generator Hostel, centrally located, with shared rooms from 25 euro per night.
http://www.cntraveler.com/galleries/2015-03-03/paris-france-newest-hotel-generator-hostel
http://generatorhostels.com/en/destinations/paris/#

Another vote for skipping the Champs-Elysées. You'll see as much as you need to see of it from atop the Arc de Triomphe. Zip on the metro down to Concorde and save your walking energy for another segment.

Thanks tremendously for your comments--I hope to follow up on all your suggestions and finalize plans in the next few days.

Do you want to pick a couple of museums and see them well? Maybe pick a theme? Or see highlights? Or focus on an artist? Or style? Don't forget to factor in some down time for cafes and people watching. IMO you can skip Champ Elysees unless you are into shopping. You need a full day for Versailles, and it is great, but exhausting. To stick to your budget sounds like some time needed for shopping for picnics and finding picnic spots by the gardens and river. Are you planning to go up the Eiffel tower? If so, get your ticket on line. Also, look for free entertainment--music, street art, people watching.Supposed to be great views of Eiffel Tower from Trocadero, esp. when lit at night. Good luck with your planning. I am revamping plans too. My list is always bigger than the time allows.

Thanks for all the advice, Paris was lovely. Followed your advice for the most part. I squeezed in the Champs Elysees & Arc de Triomphe, Sacre Coeur & Montmartre, and the Eiffel Tower the afternoon / night I first arrived. I returned to the Eiffiel to climb up it for an unforgettable view of the region after visiting Norte Dame, Luxembourg Gardens Tuesday. Did the 2 day Museum Pass. Went to several museums including the (Orangerie, St. Chapelle, Musee Rodin etc.) Louvre on Wed. Finally I took a train to and from Versailles before checking out Invalides and the Orsay Museum on Thurs. I saw much of the city--totally destroying my shoes in the process lol!

Sounds exhausting but so very rewarding! Thanks for sharing your outcome! If you want to go into more depth, there's a "trip reports" section on this board. Others might learn from your experiences, positive and (if any) negative.

You really hit the ground running (hence the worn shoes). So glad you had a great time.