Does cardiac output vary with metabolic rate? If metabolic rate increases by 10-fold, for example, by what fa

To an extent, yes. If metabolic rate goes up, then oxygen delivery must go up. This can be done by a couple ways. First, the cardiac output can go up. But, even when the cardiac output doesn't go up by enough, then the percentage of oxygen extracted from the blood increases. So, the oxygen saturation of mixed venous blood in a regular person might be around 70% or so, but may drop to reflect the increase in oxygen consumption. I'm not sure, though, if there is a fixed way to predict how the body will respond. I'm sure it also depends on things like how the metabolic rate increases (stress v. infection v. exercise) etc.