I take it you're coming from the UK, so right, you won't have any visa hassles. You also won't have any carte de séjour problems either. The carte de séjour is no longer issued to EU nationals in France.

OK - to your questions:
Rental on a one-bedroom flat can be literally just about anything. It depends on the area. Rural areas are considerably cheaper than cities. There are a few magazines that cover rental and house purchasing ads for all France that might give you an idea about various areas (and give your French a bit of practice before you get here). The granddaddy of them all is Particulier à Particulier here: IMMOBILIER DE PARTICULIER A PARTICULIER. Choix entre des milliers d'annonces immobilières de particuliers, location, achat et vente en immobilier | PAP.fr Click on "Location" for rental listings.

English tutors are generally pretty sought after by folks anxious that their kids should learn English, though I wouldn't count on the pay being too extraordinary. You'll also have to set yourself up as a business and get yourself registered for social insurances. It might be easier to try and find a language school to hire you at first, get you your social security number, and then once you've got a bit of experience under your belt, strike out on your own.

The language schools tend to have the corporate accounts sewn up, and its the corporate accounts that tend to pay the best when it comes to language training.

The best site I know of for all the legal stuff you'll need is SERVICE PUBLIC - Particuliers - but again, it's in French. The section on Etrangers in France is actually quite good, and you'll want to check the section on Emploi. As with most information in France, your mileage may vary. Different towns and regions tend to interpret the rules in their own way, but this site gives what are supposed to be the official rules.
Cheers,
Bev