Learn Danish: Vi har inviteret alle gæsterne undtagen svogeren. - We have invited all the guests except the brother-in-law.

In this sentence, they all mean the same thing and are grammatically interchangeable. You could just as easily say bortset fra svogeren or på nær svogeren. However, undtagen is highly common and natural when directly following absolute words like alle, ingen, or alt.

Can I move the exclusion to the beginning and say Undtagen svogeren har vi inviteret alle gæsterne?

It sounds very unnatural to start a sentence with undtagen. If you want to place the exception at the beginning for emphasis, you should use bortset fra instead. Undtagen is almost always placed after the main statement.

Why is it alle gæsterne with the definite ending, rather than just alle gæster?

When you mean 'all of the' within a specific group, Danish uses alle plus the definite noun (gæsterne). If you said alle gæster, it would mean 'all guests' everywhere in a general sense. Since you are talking about the specific guests for your event, you need the definite suffix.

Does undtagen change its ending depending on the noun that follows it, like an adjective would?

No, undtagen is invariable. Even though it historically comes from a verb form meaning 'excepted' or 'taken away', it functions as a fixed preposition or conjunction in modern Danish and never changes to match gender or number.