In reply to McDuck: Going out in full Winter conditions, I'd ALWAYS carry a short length of rope plus a sling and HMS carabiner.
Even without harnesses, that is then enough to rig a buried horizontal axe and bucket seat in order to belay a leader over a cornice or to use an Italian hitch to belay the party up or down steep or exposed steps on ridges. I'd say that would be enough for all grade Is, most grade I/II routes and some grade IIs (i.e. gullies under deep consolidated snow or routes where the difficulties are known to be short or avoidable).
Equally, there are other grade IIs where I'd carry no additional gear than perhaps a second sling & krab and some abseil tat, but I'd always go with harnesses even though there would probably be a fair chance of the rope staying in the rucsac. Routes like the Anoach Eagach would fall into that category as they can vary massively in condition from a pleasant walk with 3-4 short scrambles with good handholds on dry rock to a full winter expedition covered in deep hoar.
Finally, there are other grade IIs that feel more like proper climbs especially in lean conditions (The Runnel, Spiral Gully) where a few wires might not go amiss. From grade II/III upwards and certainly for all grade IIIs I'd much prefer to approach them roped up and carrying some sort of lead rack, even if it is not subsequently used.
As regards footwear, crampons and axe/axes, things are less clear cut. I would very happily solo Tower Ridge (give grade IV) with exactly the set-up you have described but there are a fair few grade II gullies I'd stay well clear of without both a second axe and stiffer boots (and an ice csrew or two). As a rough guide, a single axe and walking boots/crampons (rather than mountaineering boots) are not a great handicap on most ridges and easier mixed routes (e.g. Dorsal Arete, grade II) but you will rapidly struggle on routes with short icy steps.