Entropy Rate of Bit String How to calculate "Entropy Rate" of a bit string?
Does "NIST Randomness Test Suite" provide any test to measure Entropy of an 
entropy source?
How to check "biased-ness" of a bit string?


Thank you in advance.
There is no such thing as the "Entropy Rate" of a bit string. That's shorthand for the entropy rate of the process that created the bitstring. Thus what's to be analysed to determine this entropy rate is how the bit string was produced, not the bitstring itself.

If a bitstring 0000 was produced by four fair coin throws, the entropy rate of that process is 4 bit/bitstring, and the entropy rate of the process that created each bit of the bitstring is 1 (bit/bit). Knowing the bitstring is of no use in this determination.

If a bitstring 0101 was produced by starting with 0 then alternating, then the entropy rate is zero.
The "NIST Randomness Test Suite" accepts a bitstring as input and determines if the hypothesis that the bits in this bitstring have been created independently and uniformly at random is acceptable, or should be rejected as too improbable.


When the hypothesis is true, the tests most often accept the hypothesis, but occasionally reject it (modern versions output a p-value, that is the probability that the hypothesis is rejected even though it actually holds).
When the hypothesis is false, what the test concludes depends enormously on the actual process.


If the process is a CSPRNG initialized with a constant bitstring, its entropy rate is zero, but the tests conclude just like if the hypothesis was true (and the entropy rate was 1 bit/bit).
If the process is a natural source that behaves distinguishably from the hypothesis (for example, if each bit is determined by 20 fair coin throws and is 1 if there was at least 10 heads; which is a source of independent bits sizably biased towards 1), some tests in the suite are more likely to reject than they would be should the hypothesis hold; how likely depends on the source, and on which test.


To check the "biased-ness" of a bit string, first define "biased-ness" of a bit string; I know no acknowledged definition.

To test if a bitstring could be from a source of random independent bits with no bias, count how many bits it has, and how many are 1; then, armed with these two integers, use a (perhaps, one sided) chi-squared test. This will tell if the hypothesis can be rejected with confidence. The monobit test in the NIST Randomness Test Suite does that, restricted to long bitstrings.