RubeRad

Another vote for this path. Set a search for mountain bikes, capped at $200 or so, look at bikes with rigid forks. Since you are not interested in learning how do do your own wrenching, budget like $100-200 for
* shop tune-up ($50-75)
* slick tires (26x1.75 or so)
* decent lock
* maybe/eventually a rack, meanwhile backpack
* maybe/eventually lights, meanwhile daytime only

For budget lights, my recommendation is a zoomable-head cree xml-t6 flashlight that takes a single, rechargeable 18650 battery (or 3 AAAs). Like here's one (including battery and charger) for $10. Attach it extremely securely to your bars with two interlocked hose clamps, tightened down with a wrench. Expect to upgrade fairly soon to a decent battery from an american supplier (one option: batteryjunction.com). Tail lights are only "be seen", not "see by", so cheap options are fine.

I got no ideas for you wrt fenders/rain, I live in San Diego . Check that, one idea, instead of expensive cycle-branded rainwear, consider an Ikea Knalla poncho. And if you are really immune to embarassment (or proud of homebrew solutions, which it seems you might be), tie on plastic shopping bags as disposable shoe covers. (I guess that makes two ideas then...)

Minimal maintenance, you need to know how to repair a flat on the road, I recommend Pedro's tire levers and the Park VP-1 patch kit (don't waste your money on 'glueless' patches, they're just stickers and you'll be lucky if they get you home). You'll also need a portable pump, cheap ones are good enough, and it can serve as your home pump as well, until you get to hankering for a nice stand pump with a pressure gauge.

Lube your chain if it starts to squeak. As a low-maintenance guy with no desire for a showroom-shiny bike, I would recommend Chain-L.

Beyond that, you should be able to Just Ride.