Absolutely! Stylish sneakers are acceptable in Europe and you'll see many Europeans wearing them as they go about their daily lives.
Dress codes in Europe tend to come in two types: posh restaurants and religious sites.
I generally only bring one coat, while I use for style, warmth, and rain protection, as needed.
Your purse for your week in Europe should have a few attributes:
-Small
-Light-weight
-Hands-free
-Not easily taken/opened
You certainly can wear jeans in Europe, and for a week-long trip, a pair of jeans is, frankly, a must-have. Keep your jeans darker and avoid denim with holes or bleaching.
It's a good idea to tone down the athleisure by not wearing leggings as pants, mixing in non-athletic materials (like a denim jacket), and keeping logos and bright colors to a bare minimum.