+1 for mud guards and disk brakes.
Pinnacle Arkose looks good.
Drops Vs flats is personal preferance, but I’d rather drops over any distance on road, they’re more comfortable.
13miles isn’t a killer, but it is enough to tire you out if you do it every day. On the other hand it’s more than most on here probably do in ‘training’, so you’ll get fit very quickly. I’d take a day off and get the tube once a week with clothes, towels, etc.
Budget for decent clothes and kit too. DHB from wiggle is hard to go wrong with. But a wardrobe of winter kit is never cheap. Even a basic roubaix tights, jersey, softshell/fleace and a waterproof will be £200+, plus shoes, gloves, helmet. No point having anice bike buit it being too cold/wet to ride it for the sake of decent clothes.