If you need to use GOT, then the answer would be: It seems that Tony GOT LOST on his way. By the way, my name is Tony and I rarely get lost. :)

Yes, it is correct, and I agree with Tony and Shawn. This sentence only works in British English, where the past participle of 'get' is the same as the past - 'got'. The answer is "Tony seems TO HAVE GOT LOST on his way" It doesn't work in US English, but don't worry - there isn't a mistake in your book. I also agree that 'while coming' and 'on his way' are a little unnecessary and unnatural, but sometimes test writers are forced to write some odd-sounding things to make the questions work. By the way, I'm not only a native British English speaker, but I also write these tests as a profession, so you can take it from me - the sentence is fine.

The first sentence should be "It looks as though Tony has lost his way while coming." or "It looks as if Tony has lost his way while coming." in American English. Most Americans are just going to say "It looks as though Tony has lost his way." though since the "while coming" is kind of obvious and not needed. The second sentence is "Tony seems to have gotten lost on his way." in American English. When it is clear from context that he is on his way, then you can just shorten it to "Tony seems to have gotten lost." You can also say, "Tony seems to have lost his way."