I do a slightly shorter commute.
For a short commute like that, just wear your work clothes rather than messing about getting changed. Ride a bit slower on your way in to avoid sweatiness.
Bike: I’ve ridden everything from a cross bike to a cargo bike, via a MTB and a sit up and beg German bike. The cargo bike is the most sensible, the cross bike is fastest and fun.
Lights: Two front, two back. If one dies, the other will still work. I have a Cateye set, plus a cheap set from Aldi – the cheap ones just stay on the bike as it’s left somewhere reasonable secure at work.
Bag: I’ve used a rucksack, a basket, panniers, courier bag. My favourite is a ‘basket’ on a frame-mounted front rack.
Helmet: I do, but mainly to stop my mother-in-law from nagging me.
Jacket: currently using a cheap Aldi one. Commuting ruins things, so go cheap, if possible.
Waterproof trousers: again, cheap Aldi.
Lock: heavy D-lock that lives at work at the moment, as I don’t bother stopping on the way home.