wrt your first question: this depends on your software of choice. There are really two types of p-values that are used frequently in these scenarios, both typically based upon likelihood ratio tests (there are others but these are typically equivalent or at least differ little in their results).
It is important to realize that all of these p-values are conditional on (part of) the rest of the parameters. That means: Assuming (some of) the other parameter estimates are correct, you test whether or not the coefficient for a parameter is zero. Typically, the null hypothesis for these tests is that the coefficient is zero, so if you have a small p-value, it means (conditionally on the value of the other coefficients) that the coefficient itself is unlikely to be zero.
Type I tests test for the zeroness of each coefficient conditionally on the value of the coefficients that come before it in the model (left to right). Type III tests (marginal tests), test for the zeroness of each coefficient conditional on the value of all other coefficients.
Different tools present different p-values as the default, although typically you have ways of obtaining both. If you don't have a reason outside of statistics to include the parameters in some order, you will generally be interested in the type III test results.
Finally (relating more to your last question), with a likelihood ratio test you can always create a test for any set of coefficients conditional on the rest. This is the way to go if you want to test for multiple coefficients being zero at the same time (otherwise you run into some nasty multiple testing issues).