I'm so excited for you!!!
I've never traveled to London in the same way you're planning too, but I did work as a Group Leader for an international summer school company and toured groups of high school kids around Europe! I've never been to Luton, but I was able to find this info for you about the shuttle/train service:
http://www.londontoolkit.com/travel/luton_rail.htm
Croydon is only about 15 min away from London by train, but if you're planning to do most/all of your sightseeing in London, it may be a pain (and a bit expensive) to go back and forth all the time. Sometimes there are spare rooms you can rent in university housing that are worth a look (I'm pretty sure LSE does), but if you find that it's still going to be cheaper in Croydon and going back and forth, it's not all that bad
With only 4 or 5 days in London and Paris, the best thing to do would be to decide on your "must-sees" and go from there. Note that a lot of the tours and admission costs are pretty steep, but just going to look doesn't cost anything!
I really loved seeing Big Ben, Westminster Abbey, Picadilly Circus, Oxford Circus, and Buckingham Palace - I didn't do the London Tower or Madame Tussaud's, because I didn't feel like paying for them, but I hear they're fun to visit, especially if you can get a discounted multiple visit pass to several attractions. It is AMAZING to see Shakespeare performed at the Globe, if you can get tickets! There are also lots of inexpensive shows you can see in the Picadilly Circus area. Stonehenge and Oxford are a little out of the way, but are totally worth the trip. If you'd like an awesome tour of the city, go on a double decker bus tour that you can hop on and off at each destination. If you're a Harry Potter fan, King's Cross station is easy to get to, especially if you're already on the Tube to get somewhere.
Paris is my favourite!
Must-sees include the Eiffel Tower, the Louvre, l'Arc de Triomphe and les Champs Elysees, Sacre Coeur, la Musee d'Orsay, Notre Dame and the Latin Quarter, and some awesome day trips include Versailles, Fontainebleau, Chartres, Vimy Ridge and Giverny. There's some awesome shopping to be done around Chatelet-Les Halles and St. Germain. Les bateaux mouches offer nightly cruises - aim for one around 8:30 in front of the Eiffel Tower to see it lit up at night!
In terms of traveling, never let a stranger handle your suitcase, be wary of people offering you items in the streets (bracelets at Sacre Coeur are notorious, as is an engagement ring ploy along les Champs Elysees). Don't carry too much cash on you, and if at all possible keep it close to your body and hidden in a traveler's pouch, just to be safe. Also, while it's tempting to pack your days with stuff to do, make sure you pace yourself - take the time to enjoy the scenery, otherwise you'll be too stressed and exhausted to enjoy any of it!
My absolute favourite, low cost thing to do in Paris? Grab a fresh baguette, some cheese and a bottle of wine and picnic in the park - under the Eiffel Tower is an awesome spot, as is the tiny park across from Notre Dame in the Latin Quarter.
Woo, sorry, I didn't realize I had so much to say!! I hope you enjoy your trip - I'm insanely jealous, I miss Europe!!