Learn Turkish: Karısı doktor muymuş?

The -muş here shows that the speaker is asking about information they heard or learned from someone else. It is like asking, 'Did you hear if his wife is a doctor?' or 'Is it true what they say, that his wife is a doctor?' English often just uses the present tense 'Is...' for this, but Turkish specifies the secondhand source of the information.

What is the y doing in muymuş?

It is a buffer letter! The reported past suffix -miş was originally a separate word (imiş). When it attaches directly to a word ending in a vowel, like the question particle mu, the i drops and a y is inserted to keep the two vowels from clashing. Following 4-way vowel harmony, mu + imiş becomes muymuş.

How would the meaning change if I just asked Karısı doktor mu?

Karısı doktor mu? is a simple, direct question: 'Is his wife a doctor?' You would ask this if you just want to know a straightforward fact. By adding -muş to make Karısı doktor muymuş?, you are specifically asking about rumors, secondhand information, or confirming something you were told.

Let's review karısı. Which suffix is attached to karı?