St Remy is lovely. Hire an electric bike to see the nearby villages and countryside, no need for car. Definitely no car required in Lyon, set aside time and money for fantastic meals in Lyon. Arles, Avignon, these are classic tourists spots for a reason! Well worth seeing.
Eurostar is not operating from Lyon to London. Eurostar is the best choice to travel from central Paris to central London for comfort and speed. From Lyon to Paris, should you wish to have time in that city, take the TGV. As mentioned above, pre purchase as soon as the time allows for obtaining the best fares (which is now for late October). I once heard, unsure if true, fares are cheaper on the french eurostar website than the UK eurostar website.
Best idea is to borrow a couple of guide books from your local library. Guide books I feel piece together better itineraries than stand alone articles. Whichever guide you then like best go and buy. Books don't have roaming charges:)