You can't know exactly how much noise your ciphertext has because when you encrypt a message, you sample random values that are combined to the message and they also make part of the noise (so, the noise is also random). But you can have good estimations for the magnitude of the noise because all the distributions from where we sample the values are known.
That said, the better you can do to keep track of the noise growth is to use the estimative provided (generally) by the authors of the scheme: in general, for the scheme we are using, we know upper bounds to the noise of a ciphertext that is the result of an operation, that is, we know functions $f$ and $g$ such that
if $noise(c_1) \le n_1$ and $noise(c_2) \le n_2$, then
$noise(c_1 + c_2) \le f(n_1, n_2)$ and $noise(c_1 \cdot c_2) \le g(n_1, n_2)$
(For instance, we could have $f(n_1, n_2) = n_1 + n_2 + 128$ and $g(n_1, n_2) = 128 \cdot n_1 \cdot n_2$).
It is worth noting that those estimations are quite pessimist: for instance, on the paper A Comparison of the Homomorphic Encryption Schemes FV and YASHE, authors say that they could select parameters with size about 30% smaller than expected because the noise growth was always below the theoretical upper bound, as they show in the following figure (which is on the 12th page):
That difference occurs because the theoretical upper bound uses a kind of "worst-case analysis", considering the biggest noise that can occur, but in practice, that biggest values of noise will occur only with small probability (remember that, in general, we sample those values following a Gaussian, therefore values close to the mean are more likely to occur...)
- Is it even possible to clip off the unwanted noise and continue with your operation?
Yes. This is called bootstrapping and it consists in running a "refresh" procedure to remove the noise of the ciphertext. But that is not a simple procedure and can be quite expensive computational-wise.
- Now by learning that noise growth what else can I gain?
You can try to reduce the number of times you use the "refresh" procedure removing the noise only when it is needed instead of doing that after each operation.