Thermal paste only really does ONE thing, fill in the cracks on both surfaces to make them smooth, thus providing better heat transfer. I have ran a computer before, an older Sempron system with a 130nm CPU can't remember the exact spec anymore, with out thermal paste on a stock heatsink, got warm, but it never had issues. But you should invest in some sort of thermal paste.
Depends on the CPU I guess. I'd imagine it'd be fine for a little bit under low load situations, with cool-running CPUs. Just don't try it with no heatsink at all.