Windy, that was wonderful. Thank you.
lucianog - You need a boat that draws less than around 1.5m and is either small enough that you can easily drop the mast or has a deck stepped mast set in a tabernacle. Alternative t the mast in tabernacle is to have the mast pulled as you enter the canal system , strap it on deck and then re step it at the other end. This of course precludes you from doing any sailing along the way but if you are simply going from the Channel to the Med then sailing is not really a practical option.
I'm afraid I can't help you with the shortest time it would take but there are plenty of web sites with information on the canals. This is one of the best I have seen.
Cost ? About the same as a length of string.
Ref timing.....you do not want to hurry. The trip would have to be one of the greats, if only for the food and the wine.