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What is the meaning of the following sentence(Source: The Man Who Liked Dogs by Raymond Chandler)

I looked at a small alcove office with a roll top desk and a waiting
room with mission leather chairs and three diplomas on the wall,
at a mission table scattered with copies of the Dog Fancier's Gazette.

?

Does it mean "I looked at a small alcove office with a roll top desk.
And I looked at a waiting room with mission leather chairs and three diplomas on the wall.
And I looked at a mission table scattered with copies of the Dog Fancier's Gazette" ?

Can we remove the "at" from the sentence? (=> I looked at a small alcove office with a roll top desk and a waiting room with mission leather chairs and three diplomas on the wall, a mission table scattered with copies of the Dog Fancier's Gazette.)

I think your understanding of the basic meaning is right. Also your intuition seems valid that this passage doesn't really need both "at"s.

Not that I have the stature to criticize Mr. Chandler's writing, but I am a little puzzled why he put the "and" in the middle of the list of things he was looking at. It makes it little bit tricky to visualize the various furnishings in the office and/or the waiting room.

I have a feeling the rest of the story will probably make sense without understanding this sentence perfectly though.

Does it mean "I looked at a small alcove office with a roll top desk. And I looked at a waiting room with mission leather chairs and three diplomas on the wall. And I looked at a mission table scattered with copies of the Dog Fancier's Gazette" ?

Yes.

Normally the two participial phrases would be joined by and as you've guessed. The omission is a literary device called asyndeton, which here is meant to create an atmosphere of action and realism.

The idea is that the subject matter is harsh realities, so flowery prose would be incongruous—either cruelly indifferent or idiotically oblivious. Terse narration signals awareness and even empathy.

You'll see the effect repeats throughout the story:

He looked around and under me, [but he] didn't see a dog.