Can a fixed gear sprocket be changed without tools?
For the lockring a hammer and screwdriver works sometimes, but you end up damaging everything involved except the hammer, sometimes to the point that you can't get the lockring tool to work after the fact. 
Get the lockring tool. There are semi universal ones that work ok. The type that are designed to fit specific lockrings (ie 3 slot, 4 slot) work much better. Either way they're fairly cheap. Once the lockring is off you might be able to use your bikes chain and crank in place of a chainwhip, but this is yet another place where you're better off just buying the appropriate tool. Once you have them you'll be a happier fixie rider.