I wonder if we have the problem of pilot contamination in LTE especially in the case of TDD, as the users also need to send the pilots to the base station, so we still need a large number of orthogonal pilots as the case of Massive MIMO TDD.
Yes there is definitely pilot contamination in LTE TDD, because of the problem you mentioned (non orthogonal pilots). Attaching an article from Dr. Gabor Fodor of Ericsson Sweden (whom I met in ICC London) on pilot contamination and mitigation in LTE. This should help.
Yes, there is and there is also pilot contamination in FDD systems. But this is not a big issue since the systems are relatively lightly loaded. In most cases only 0, 1 or 2 active users per cell at each time instant. And if there are more users we can orthogonalixe them by OFDMA. It is more of a challenge if there are 20-40 active users per cell and all use the entire frequency band, as in a massive MIMO system.