Help in interpreting the auto correlation graph
In my opinion this question is too broad. The first point is that you want to see if a given time series is random or not. I have doubt on how we can define this term here. Also you don't mention any distribution. Normally, we are interested to see if a time-series belongs to a specific distribution or not, for which there are standard techniques. 

Regarding dependency, you should note that correlation only captures linear dependencies. You can have random variables that are highly dependent but have very low correlation. So it is important that what specific type of dependency you are looking for.

Edit: That said, referring to the ACF a peak at lag zero is visible. The reason is that at zero lag the time-series shows maximum similarity which is not surprising since it is obviously similar to itself! Note that the amplitude of this peak for a general complex signal $x[n]$ is $$R_{XX}[0]=\sum{x[n]x^*[n]}=\sum{|x[n]|^2}$$ which is the signal's power. You can easily show (theoretically) that $$R_{XX}[k]\le R_{XX}[0], \,\forall k$$

In addition, you should note that the autocorrelation of a white process (i.e. a process with uncorrelated random variables) is a pure delta function. So closer your autocorrelation to a delta means it is more uncorrelated, which is a good thing for a PRNG.

Regarding, your question about mutual information and entropy, yes both of them are used. Entropy is a well known measure for randomness. I cannot give you more details in here but can suggest you looking at this or similar resources.