If you disconnect the battery it will indeed lose the fuel trim numbers and will need to be relearned, it won't relearn them all until you go between every load and rpm combo in closed loop. As a guide I've always taken a tank of fuel as enough to say it's relearned back to where it should be. Having said that it shouldn't make a huge difference to the fuel useage unless there was something wrong and it was compensating with fueling (such as an air/vacuum leak).
In cold weather it will use more fuel as the air is denser so adds more fuel, conversely you will get slightly more power from it so for overall engine running there won't be a huge difference (unless you have a lead foot). There will be a difference in loads from extra electrical features used and warm up times compared to summer but again it shouldn't be huge unless it's left to sit and idle to warm up (I don't think the dash computer takes into account stationary fuel useage).
So you will use slightly more fuel in winter compared to summer but not drastically so (about 10% more in winter depending on your driving style).
I assume you are getting your mpg figures from the dash computer? Was your 11mpg doing exactly the same roads, in exactly the same conditions and over the same time period you got your 22mpg from?
Unless you are comparing like for like then I would take what it says with a pinch of salt and even then I wouldn't rely on it heavily.
Mine says it does 20-24mpg on the dash info but it doesn't, it does 17mpg from tank to tank fill which is bang on the manufacturers quoted average which I'm happy with given the driving I do. I have slightly undersize off road tyres hence the difference between the dash and reality. I could change the correction factor numbers (in the hidden dash menu) and get it to display whatever mpg I wanted to be honest but I just treat it as a guide but only if it's comparing like for like.
In your case I would run it for a tank full or two, working the mpg out from fill ups rather than the dash and then see what it's doing. If it's still using a lot of fuel after that then get the codes read and study the live data fuel trims.
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