For winter walking a traditional straight-shafted mountaineering axe seems most appropriate. This would include products like the very common BD Raven or Raven Pro, and also those with a slight bend such as the Petzl Summit, Grivel Air Tech Evo, and BD Venom.
As there are existing questions regarding length (How do I know what size ice axe I should get?), I'll be brief and only repeat the traditional advice of holding it by the head with your arm hanging down and picking a length where the spike reaches your ankle. (E.g. I'm 185cm tall and use a 70cm axe. I would probably be happy with a 65cm length as well. Others may advise shorter lengths; this becomes more applicable as the slopes become steeper or you don't expect to use the axe for much of the route.)
These days nearly all mountaineering axes have very similar features: a positively-curved steel head and adze, a steel spike, and a straight or slightly curved shaft. A few products deviate from this: Grivel's Futura replaced the adze with a plastic handle and Petzl's Sum'tec has a pick with negative curvature. Ultralight models may use aluminum for the head and eliminate the spike, but these are not suited for sustained use or harder snow and ice.
A negative curvature pick may be useful on technical routes, but is not so good for self-arrest. A straight pick is somewhat old-fashioned and not commonly seen. Note that features on the shaft (finger rests, rubber handles, etc.) may make it more difficult to plunge the shaft. With modern crampons the adze is less used for chopping steps, although it's still quite useful if you need to carve out a platform or snow bollard.
It's not until you get into dedicated ice tools that you run into significant differences: interchangeable picks, being able to select between a hammer, adze, or neither, significantly curved shafts, and very ergonomic handles.
I personally don't use a leash, although some people do. Regardless of your choice, there are two very important rules to keep in mind. The first is Do Not Fall, and the second is Don't Let Go Of The Axe.
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