Learn Turkish: Bu adam kötü değil.

In Turkish, the negation word değil always comes after the noun or adjective you want to negate. So instead of saying "not bad" like in English, you say kötü değil (literally "bad not").

Where is the word for "is" in this sentence?

Turkish doesn't use a separate word for "is." Normally, "am/is/are" is added as a suffix to the end of the sentence. However, for "he", "she", or "it" (third person), the suffix is usually left completely blank. Because bu adam (this man) is a "he", değil needs no extra ending.

Could I use hayır instead of değil here?

No, hayır strictly means "no" (like when answering a yes/no question). When you want to say that a noun or adjective is not something, you must use değil.

Does bu change if the noun is different?

No, Turkish doesn't have grammatical gender, and adjectives or pointer words like bu (this) never change to match the noun. It will always be bu, no matter what noun follows it.