I have often stayed with friends, although perhaps not for as long as a week. But… I barely know you, so I’d feel a little weird about it. Let’s assume it’s someone I know a bit better than I know you.
I’ll bring a house-gift, unless you are family or the friend I stay with all the time.
My drink of choice is tap water, unless your water is horrible, in which case I hope you keep bottled water around. I would feel comfortable pouring myself a glass of water, and I’d put the glass in the dishwasher, or wash it and leave it in the rack if that’s how you roll. I wouldn’t feel comfortable taking any food items you didn’t explicitly invite me to take. So, if you opened the pantry and said, here’s the breakfast cereal, feel free to have some if you get up before us", I would, indeed, feel comfortable taking it. But if you didn’t invite me to do so, I wouldn’t. I’d sneak out to a diner and grab something.
Hosts usually do show me the tea and the tea-making equipment (or coffee, etc.) and I will use that if I’m up before them in the morning. If I have reason to believe they will want coffee, too, I will make enough for them. I don’t make them tea, because I tend to make tea stronger than others like, and I’ll misjudge.
I would certainly look at your artwork, and your knicknacks, and your books. And I would likely make small talk about them. I also might peek behind closed doors. I’m nosy. But I wouldn’t touch anything or disturb anything.
Yes, I’d certainly take a walk in the neighborhood, and I’d knock when I got home (or call out “hello”) unless I thought you might be asleep. Yes, I’d ask to use the washer if I ran low on underwear, although I’d plan so as not to need to, unless visiting you was part of a long trip But I wouldn’t feel weird about asking, and I’d be surprised if you didn’t give me access to the washer, and any relevant instructions.
I’ve never scrubbed the shower. But when I visited a friend who had a squeegee in the shower, I carefully squeegeed down the walls after every shower. And I will keep my mess contained to the room you put me in, or to my suitcase, if you put me in the living room. The bathroom I used will look tidy. I won’t use more than one towel, either. I’d use two in a hotel (one for my hair, another for my body) but I don’t want to leave two damp towels lying around. I dunno, I like to keep a low profile as a guest, and not take up too much space.
I will take you out to dinner at a nice place while I’m there. I will enthusiastically eat the food you put in front of me, too. If you are cooking while I’m around, I will ask if you’d like help, and I may end up slicing the vegetables or something. Once, a friend had me picking herbs and fruit in her garden before preparing it. that was fun. 
I’ll clean any dishes I might have used on my own, and I’ll wash some pots or empty the dishwasher if that seems helpful.
If I’m doing fun sightseeing, I will invite you along, my treat, but understand if you have other stuff you are doing.
At the end of my stay, I will strip the bed, and fold up the dirty sheets and towels and leave you a tidy pile.