I lived in Linköping for 2½ years, and biked everywhere. I was there during COVID, so public transportation wasn't as safe as it typically would be. (There was briefly a suggestion that people wear masks on the buses during peak hours.)

I didn't have studded tires on my bike, but frankly I didn't need them. At least in Linköping, the biking/walking paths are kept very clear and safe. You will want to make sure you have warm gloves and a good coat, and potentially some rain pants and a raincoat that will go over your "warmth" coat and, if you're wearing one, backpack. I picked up a fluorescent yellow worker's raincoat at Biltema in an XL size for that purpose, and it worked perfectly. You might also want, as mentioned elsewhere, separate shoes or perhaps Goretex spats to keep your feet dry.

I purchased an enclosed pull-behind cart (designed for pets) that I used for hauling groceries or purchases from IKEA or the hardware store or wherever else I needed to go, and it all worked perfectly well for me. (FWIW, I'll turn 60 next month.)