Need help in understanding minimum variance estimator and CRLB concept using an example
You may want an estimator with minimun variance because each time you calculate it, it is likely to be closer to its expectation, and in the case that the estimator is unbiased, closer to the true parameter. An estimator is inconsisten if increasing the number of samples of the dataset to calculate it doesn't reduces its variance. 

The sample mean estimator of X:
$$
\bar X = \frac{1}{N}\sum_{i=0}^{N-1}x_i
$$
is unbiased and its variance is:
$$
\mathbb var(X)=\frac{\sigma^2}{N}
$$
It is clear that as N increases, the variance gets smaller, so it is consistent.

Regarding question 1, NO. For instance, one of the two estimators can have a higher CRLB always, for both high or low SNR. The criteria for choosing an estimator depends on the problem, the prior knowledge about it, the computation constraints and so on. Normally it is desired a minumn variance and a non-biased estimator, i.e. an efficient estimator. But, if your decision is only based on the CRLB, then choose that one with the smaller one.

Take this 'silly' example: for a dataset you decide that the estimator for the mean is always (arbitrarily) 3.5 even if you know nothing about the data and you do no calculations... It is clear that this estimator has variance 0, because it will be always 3.5, but it is strongly biased, it is randomly biased indeed... is it a good estimator? No, although it has 0 variance. The key point here is that yo can't have an unbiased estimator with a lower variance than its CRLB (if the CRLB exists). If the variance of your estimator is lower than the CRLB, it is biased.

If you have some experience with computer simulation, Matlab or similars, it is a good exercise running Montecarlo simulations and plotting the variance and the CRLB of each realisation versus the SNR for a fixed number of samples, or/and for a fixed SNR plotting the variance and the CRLB of the estimator versus different dataset lengths.