Bayesian definitions mean that it is impossible to ever fix the probability of an event by some means external to the observer. Personally, I see some value in each stance. I think Bayesians are correct in that all observations can do is update our beliefs, and that that the idea of an objective probability is probably indefensible. On the other hand, I think it is important that we devise some standardized method by which we can all agree on the significance of a set of measurements, which is impossible if prior probabilities can be anything anyone wants them to be. Describe sets of events which are examples of each term.
Independence has two meanings.
The statistical definition is two events are independent if the probability of them occurring together is the product of the probabilities of them occurring seperately. The intuition behind the definition is that independence means there is no causal connection between the events. Successive coin flips are independent events. Mutually exclusive events are events which cannot both occur at the same time. Getting heads and getting tails on a coin flip are mutually exclusive events.