We are planning to do the exact same thing - gene (1kw) plus a small freezer - run during day to cool beers / freeze freezer blocks to put in cool bag. Freezer blocks keep everything cool at night.
As for lights, yep that works, although so as not to be too anti-social to our neighbours we run 80Ah leisure battery for some decent 12v lights at night, which is then charged by the gene during the day. We've messed around with solar panels and the like, but nothing really beats the flexibility of the gene.
Kettle, not really a goer. Even the lowest power ones are about 800w or so. The average domestic is 2-3 kw. Back to the camping stove for that one!