Is the sample proportion ($\hat p$) a random variable? Since it should vary from sample to sample, I suppose it should be a random variable. But if it is, when we write the variance of sample proportions, should we write an uppercase P-hat as the index notation of sigma?
But in most textbooks and formula tables, it seems that a lowercase p-hat is usually used



In the frequentist tradition (which is what you are using here) the random variable is the data. The population parameters are mathematically treated as constant. This is what leads to the somewhat counterintuitive "null hypothesis" setup we use in intro statistics, because the probability we return (usually in the form of a p-value) is a probability on the sample given constant population parameter set at the null hypothesis values.
I would imagine this is why you see the notation you do in introductory many textbooks.
