Honestly, I've seen bigger lava fields on the discovery Channel.
A whole bunch of Volcanoes like this could do that indeed, obscure the sun and create the equivalent of a nuclear winter. But one volcano... I am not sure.
BTW, it's impossible that anything could survive where Spock was. The temperature must have been around a thousand degrees Celsius there. For one thing, his suit should have been bright yellow from the heat! and him.. an incandescent piece of charcoal.
TBH, I'm not sure about that. One volcano could definitely be enough for that type of catastrophic event, but it would need to be a monster volcano. I'm no geologist, so I can absolutely not tell you if the volcano as depicted on screen would fit that type, but I remember it being bigger than any volcano I have ever seen in real life (the volcanoe - not the lava field. The important part is the pressure under the surface, not the size of the opening). I vaguely remember a science analysis of Into Darkness, where they said the volcano scene was mostly alright (except the fusion bomb) - but I honestly don't remember the details that good anymore.