Greetings! This is certainly an interesting question regarding the Zimmerman Telegram, and it is often glossed over in "pop-history" or tertiary sources' treatment of the document with regards to its impact on the war. Whilst it is necessary to stress here that the Zimmerman Telegram was also not the main cause for America's entry to the war, it is an interesting side-investigation to wonder why the Telegram seemed to be concerned with the Japanese as well as the Mexicans. This response shall cover both questions OP has put forward, dealing with the actual desire of the Germans first and then going onto the whole "Yellow Peril" factor. For both parts however, we must delve a tad deeper into the context behind German-Japanese and Japanese-Mexican relations throughout the war. Let's begin.

For further reading, Barbara Tuchman's rather good (if slightly outdated) book "The Zimmerman Telegram" has two entire chapters dedicated to the Yellow Peril and the Japanese factor in the entire affair, so feel free to peruse that work in the sources at the end of this response.

The Proposition: Possible Peace?

Beginning with the suggestion of Zimmerman to von Eckhardt (the German diplomat in Mexico) about the Japanese consideration, let us quote the actual bit of the Telegram itself:

"In the event of this not succeeding, we make Mexico a proposal or alliance on the following basis: make war together, make peace together, generous financial support and an understanding on our part that Mexico is to reconquer the lost territory in Texas, New Mexico, and Arizona. The settlement in detail is left to you. You will inform the President of the above most secretly as soon as the outbreak of war with the United States of America is certain and add the suggestion that he should, on his own initiative, invite Japan to immediate adherence and at the same time mediate between Japan and ourselves."

Clearly, Zimmerman wished to have the Japanese switch sides whilst the war was still going on, and after she had already shown her prowess by capturing the German port of Tsingtao back in November 1914 and maintaining a heavy naval presence in the South Pacific; and Japan had already seized many of Imperial Germany's island possessions in that area. Beyond these early engagements however, the Imperial Japanese Army found itself at (for the most part), the ends of its active belligerency in the war. Whilst elements of the Japanese Navy were kept busy in 1917 as part of an Entente Mediterranean Squadron, the Army had carried out much of its wartime involvement with great success.

It is here that we must pause the "Zimmerman Telegram" focus of the response and understand that the Anglo-Japanese Alliance was showing cracks by 1915. With the German successes at Ypres, Artois, and Champagne that same year, the Japanese government was concerned as to whether it should remain a loyal ally of the Entente Powers, or instead to enter into a German-Japanese Alliance and carry out the rest of the war alongside Germany. It was in such light that the Germans dispatched their own envoy to Peking, Paul von Hintze, to negotiate with the Japanese on behalf of the Imperial German Government. When he made his first approach to the Japanese ambassador to China, he seemed to be willing to make incredible overtures to the Japanese, as summarised below:

"Germany would let Japan keep not only Tsingtao but also the Pacific Islands, and would be willing to give Japan a much freer hand in China than would the Allies. Germany might even go so far as to finance Japan's continued expansion in China. Implied, though not expressly stated, was the idea that Japan could do as it pleased in the Far East, especially in Manchuria and Mongolia, and that Germany certainly would not restrain any such actions for the sake of Russia."

It is necessary to understand here that Japan had an...awkward alliance at best with the other Entente powers. She had entered the war to honour the 1902 Anglo-Japanese Alliance with Great Britain, but neither the British nor the Russians (especially the Russians) wanted an ascendant Japan as hegemon of the Far East. The German Foreign Office in 1915 perceptively understood this, and there essentially were willing to give Japan carte blanche in the Far East if a separate peace could be obtained. Thus through both Hintze and their embassy in Stockholm, the Foreign Office of the German Empire continued to present the Japanese with the potential benefits of a separate peace.

Here is where we ought to praise the shrewdness of the Japanese government under Okuma Shigenobu in dealing with both the German peace proposals and the fears of Allies. They made no attempts to silence these propositions, nor did they exactly keep them secret, as the Germans had hoped they would. In fact, both Hintze and the Stockholm embassy's were communicated to London and Paris shortly after their presentation, causing considerable alarm amongst the diplomats of the Entente powers. This was a stroke of genius on the part of the Japanese; if it could continue to convince both sides that it was beginning to acquiesce to the demands of the other (for the Entente, that meant accepting Germany's separate peace, and for the Germans, it was continued loyalty to the Entente), then it could dictate the terms which either side would need to pay in order to keep the Japanese on their "side" of the whole affair. Frank W. Ikle does an excellent job of phrasing this move:

"if both Germany and the Allies suspected that Japan was wavering, it would be in an excellent position to raise its terms. The greater the doubts concerning its loyalty, the higher the price Japan might command."

However, German clumsiness in attempting to simultaneously negotiate for separate peace agreements with Russia and Japan worried the Japanese government considerably. The Okuma cabinet therefore rejected German peace feelers' proposals in 1915, but through these first attempts it learnt the value of putting pressure on both Berlin and London.

German-Japanese Talks: Diplomatic Deftness

In 1916 the negotiations resumed once more, with the Germans dealing with Japanese Ambassador to Sweden Ryohei Uchida. An initial meeting between Uchida, Hugo Stinnes (a close friend of Chancellor Bethmann Hollweg), and German Ambassador to Sweden Hellmuth von Lucius took place on April 1st. The meeting made little headway, but the Germans did urge that Uchida and the Japanese government keep the talks a secret. Rather annoyingly (to the Germans), Uchida wired the results of these talks to the Japanese Foreign Minister Ichii Kikujiro, who then wired the same transmission to the Japanese embassy in London, ordering their ambassador there to transmit the news to British Foreign Secretary Sir Edward Grey. At the same time, Uchida was instructed to continue negotiations with Lucius, which occurred on the 24th of April. This meeting mostly concerned the matter of Tsingtao and its concession to the Japanese by the German government in exchange for "taking up the question of peace." Berlin was ready almost immediately to give in to this demand, and further negotiations continued.

Part 1 of 2