This is an extremely general question. Perhaps an example of the type of problem you need help with would be better.
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You could calculate it by integrating the chi-square probability distribution function but you are likely to be much better off using a table in a book or on the web.
The answer depends on what you mean by "do". Does it mean calculate individually, calculate the probability of either one or the other (or both), calculate the probability of both, calculate some function of both (for example the sum of two dice being rolled)?
Bayesian probability ; see related link .
There is no direct relationship between degrees of freedom and probability values.
The area under the pdf between two values is the probability that the random variable lies between those two values.