What is the meaning of Terry Pratchett's idiom/pun "coming and going"?

As mentioned in the comments, the "has me coming and going" part is not a pun but a normal English idiom meaning that no matter what choice you make, the outcome will be something you don't want. So it is really just a reference to the fact that Angua has two different forms available: she can be a female human or a female wolf. But the rules of the room state that no women are allowed, and that no pets are allowed, so if Angua tries to get past one rule by taking her other form, the second rule applies instead, and she is still excluded.

The additional bit about "it's only in bad taste if someone else says it" is referring to calling a werewolf (an intelligent and often highly dangerous being) a pet. As a werewolf herself, Angua can refer to werewolves however she wants; only a non-werewolf calling a werewolf a pet would be offensive.

See the idiom's definition at idioms.freedictionary.com for some more explanation.

It is a common English expression, and by that i mean its common in England, not English as in the language... Having someone "coming and going" generally means giving someone no option out, other similar expressions or sayings are "your damned if you do, and damned if you don't" and "caught between a rock and a hard place". Although the last two imply that a decision has to be made... the "coming and going" expression means basically that she loses out either way with the conditions applied. Oh and the last statement "its only bad taste if someone else says it" (referring to the "pet" clause), think of a bad racial slur that you could call yourself with no problem, If someone else calls you it... that's a different matter. Like black guys in the american movies and rap songs using the "N" word about themselves and to other black guys. if a white guy uses it to a black person, its seen as derogatory and a racial slur. She can use the term herself with no problem.