Just general observations...
Yes, you can get into a routine that you think is the "living like a local", but really, it is not, you are always an outsider, yeah, people may not care that you are there, will be polite, treat you well, but you are not one of them. (not being mean, just reality, just saying you need to behave as a guest)
You do tend to circulate in a small neighborhood, hit a limited number of shops, markets, restaurants, and that provides a familiarity, and a sense of community.
Cooking...This is a big fantasy of many, we have made some great meals, enjoy simple meals often, but to be honest, rentals rarely have much more than basic utensils and cooking vessels, and the stovetops, ovens, and other appliances are less than optimal. It takes some creativeness to take basic sauté, boiling, and bake to your expectations. Gourmet cooking everyday it is not. After a week, you will be out looking for a sharp chefs knife and a few other kitchen needs. But realistically, if you go out for three meals a day, after a week or so, you will try to find a balance between eating in vs out, more for your own health than due to cost. Maybe resorting to one meal a day out, and likely lighter than what you would have on a short trip.
As for daytrips and sightseeing, after a bit, that tapers off. You simply want to sit back and enjoy being there. Maturity is being willing just to nap the afternoon away, read a book, wander the city your in, without really having a need to "see" anything.
Basically, your doing it right if after three weeks, your in a routine sort of like you do at home. Going intense for a month or more, like you would in a two week trip, just does not work.