There are proteins (in truth, small peptides) released by neurons, but these are **not** the most typical or canonical type so I don't know where you got that information. Wikipedia has a list. Most are amino acids or derived from amino acids. They are transported into vesicles and then released when these vesicles fuse with the plasma membrane.
Neurotransmitters _have_ to be released in the axon terminal. That doesn't mean that neurons can't also release growth factors and such in the vicinity of their cell body, but the entire function of a neuron is to transmit a message from one place to another (after some processing that occurs based on received input). Neurotransmitters are the way that message is released for most neurons, and axons are the typical "output" structures of a neuron.