1) Is there a connection between the modulation kind and the channel capacity?
The capacity of a channel indicates the upper limit of how many bits can be transmitted per second over the channel with no errors (okay, technically it is "arbitrarily low number of errors", but it's basically the same thing). We do various things to try to get as close to that upper limit as we can, such as modulation types, error correcting codes, etc., but none of them affect the channel capacity itself.
The equation for channel capacity is:
$$
C=B*\log_2(1+\frac{S}{N})
$$
where $B$ is the channel bandwidth in Hz, and $\frac{S}{N}$ is the signal to noise ratio (which is unitless). Given this, the channel capacity only depends on its bandwidth and the received signal-to-noise ratio.
Going to higher order modulation types (e.g. from QPSK to 16-QAM, from 16-QAM to 256-QAM, etc.) is one way to take advantage of the $\frac{S}{N}$ portion of the equation. If you have a very high signal-to-noise ratio it is wasteful to transmit QPSK, because the constellation points are very far apart from each other. By going to 16-QAM the points are closer together, but they are still far enough apart to avoid most errors while transmitting twice as many bits. If you still have "excessive" signal-to-noise you could go to a higher order QAM signal yet, allowing more bits per symbol to be transmitted.
Why and when do we do $\log_2(M)$ where $M$ is the number of modulation symbols? For example, if the modulation is 16 QAM, then what does $\log_2(16)$ tell us?
$M$ is not the number of modulation symbols, it is the number of constellation points. What $\log_2(M)$ tells us is the number of bits that each symbol represents. There are 16 constellation points in 16-QAM, so we could number them from 0 to 15. How many bits does it take to represent 0 to 15? 4 bits, or $\log_2(16)$.
3) In equalization, when pilot symbols are send, we get better estimation quality. This is known as non-blind equalization. But in many research articles often it is mentioned that non-blind technique wastes a lot of bandwidth due to periodically sending the training symbols. What is the meaning of bandwidth wastage and how does it occur?
What they mean is that some of the bits that could have been used to transmit information are used to transmit pilot sequences instead. In LTE, for instance, in some OFDM symbols every third subcarrier is a pilot instead of data. If each subcarrier is using QPSK modulation, then each pilot subcarrier means that there are two fewer data bits.
Does capacity decreases with using training/ pilot symbols?
No, not necessarily. Yes, the number of information bits is decreased by the pilot symbols as described above, but remember that the channel capacity refers to how many error-free bits can be transmitted over the channel. The pilots are essential for error-free communication.
A similar example is error correction codes (e.g. Turbo codes). Error correction codes use parity bits or symbols to detect and correct errors. We could replace those parity bits with information bits, but that would actually reduce the number of error-free bits that we sent through, so using the error correction code is a net gain.
