Is there any other language intelligible to Mandarin Chinese speakers? Can native speakers of Mandarin Chinese readily understand speakers of neighbouring regions such as Korea, Japan, Mongolia, or Canton? Are Mandarin Chinese and any of these languages mutually intelligible?
No. Mandarin is part of the Sino-Tibetan language family, whereas Japanese, Mongolian, and Korean are not.

Even within the Sino-Tibetan family, most langauges are mutually unintelligble. Even worse, within the Chinese family, most regiolects and topolects are mutually unintelligble.

People in Shanghai, Suzhou, Ningbo speak Wu, which is not intelligble for Mandarin speaking people. The two topolects broke off some two thousand years ago.

People in Guangdong and Guangxi speak Cantonese, which is unintelligble for people speaking Wu or Mandarin. And so it continues. Only nearby topolects are intelligble, for instance Sichuan Mandarin vs. Hebei Mandarin vs. Tianjin Mandarin and so on.

Japanese has borrowed the Chinese writing system, but the spoken language has nothing in common with Chinese. Even so, Japanese has borrowed many loanwords, that often sound similar (电话 is denwa in Japanese, dianhua in Chinese). Chinese is therefore considered as the equivalent of Latin in East Asia, since many langauges have been influenced by Chinese.

At best, people from different nations can only pick up stray words in other languages.

On the other hand, they have long shared the same writing system, which means that Japanese can read Chinese and vice versa (when using characters, not Hiragana or Katakana), with some difficulty. This was true for Korean and Vietnamese before they incorporated their own writing systems.

It doesn't matter if the writing is in traditional or simplified, most can read both varieties, although may not be able to write in both scripts.