1- Yes. You need at least the shot size of the weight of the piece or pieces. Sometime, you can cheat and customize the injection patern like starting the screw rotation with the injection sequence so it will push a little more of the shot size, but doing so tend to lose some "repeativity" and do some overheat on the motlen material. With some enginering plastic, I would not try to cheat as it can damage the screw.
2- Yes, the machine can be ajust so it will only push what you need. But be aware that if you molded a 3 ounces piece on a 10 or 20 ounces shot size machine, you will lose some performance on the process.
Some problem would be a degradation of certain plastic that rest too much time in the barell due to a too big screw setup for the part. Another thing is your machine will have to heat more the barell to melt the plastic as the most heat is produce by the friction force of the plastic itself.
My advice is, if you want to make a 3ounces part and you have a 20ounces shot size machine available, why not making 6 parts at a time! Some case it could not be done but a good molder will always look to push the machine to the maximum without sacrifying the processibility or the end-product quality.