A stream cipher usually generates a stream of pseudo-random bits which get XORed with the plaintext to form the ciphertext. The stream is generated using a given IV / nonce and a secret key. CTR-mode for block ciphers is usually described as turning a block cipher into a stream cipher.
It is now widely known that the CTR-mode of operation provides IND-CPA security. Katz proves this in his book.
Does this result imply that all stream ciphers are IND-CPA?
There is a difference between the type of a cipher and the construction of a cipher. If a cipher is of a specific type for which there are known IND-CPA secure constructions then that doesn't mean that an entirely different construction is secure. There are known attacks on stream ciphers, including "modern" stream ciphers such as RC4.
A stream cipher must be used correctly for the encryption scheme to be IND-CPA secure.
If the generated key stream is secure then the XOR of the plaintext with the key stream will of course also be secure (up to the specified limits of the stream cipher anyway).
No. Indeed, as in the answer by Maarten, it depends on the security and strength of the stream cipher. However, even if the stream cipher is a secure pseudorandom generator (which is its proper modeling), encryption is not necessarily CPA-secure when XORing the pad with the plaintext. This is also explained in great detail in Katz-Lindell.
In fact, it is only guaranteed to be eavesdropping-secure due to the problem of pad reuse (for a real attack that used this, see this paper - The Misuse of RC4 in Microsoft Word and Excel).