Authenticating small packets
No, in general using a CRC in this way is not secure. A CRC is not designed to be used against adversaries, it is used to detect random bit changes to the data it is protecting (as well as the CRC itself). A CRC of 16 bit will certainly not be as secure as an 8 byte MAC value, that was designed to protect against such attacks. Without additional measures a 16 bit CRC would be pretty easy to brute force.

Note that there are many arguments against authenticate-then-encrypt. One of the reasons that it may be less secure is that it allows attacks on the block cipher mode of operation. In this case the ciphertext may be vulnerable to padding oracle attacks.