Do demographic variables need to be standardized when population size is included as a covariate? I am doing a fixed effects regression where the dependent variable is number of people in an treatment facility in a given state/year. We are also including a population size control variable. I am wondering if number of people in treatment facilities should be standardized to per 100,000 or percentage since larger states will likely have more people in treatment facilities. Would including a population size covariate control for this? Why or why not?
 A: You incorporate the population size as an offset in the regression
For this type of count regression you would usually use the population size as an exposure variable, which is a variable that manifests in the regression as an offset with a fixed regression coefficient.  For example, suppose your response count follows a negative binomial distribution of the form:
$$Y_i \sim \text{NegBin}(\text{Mean} = \mu_i, \text{Overdispersion} = \phi),$$
using the regression equation:
$$\log (\mu_i) = \underbrace{\log(r_i)}_{\text{offset}} + \beta_0 + \sum_{j=1}^k \beta_j x_{i,j}.$$
In this formulation the value $r_i$ is the exposure variable representing the population size of the given state in the given year.  This regression equation for the model fixes the effect size pertaining to the population size so that the response count is proportionate to the population size.  (Since this is a fixed effect, there is no regression coefficient on this term.)
In this form of the count regression, you do not adjust the response variable to express it as a proportion or percentage of the population --- you leave it as a count and use the offset term to incorporate the effect of population size.  This is generally a superior approach to modelling the proportion or percentage for the response variable, since it will usually be the case that the latter will become less variable as the population size grows.  (Having said that this is superior, you will usually find that with a large amount of data, the above negative binomial regression gets similar results to a Gaussian linear regression on the logarithm of the proportion-based response, so long as you adjust the error variance to incorporate the resulting heteroskedasticity.)

Testing the effect of the population size: In some case, you might hypothesise that the response variable is affected by the population size in a way that is non-proportionate.  In this case you can add an additional regression term representing the additional effect of the population size, giving the alternative regression equation:
$$\log (\mu_i) = \underbrace{\log(r_i)}_{\text{offset}} + \underbrace{\alpha \log(r_i)}_{\text{population effect}} + \beta_0 + \sum_{j=1}^k \beta_j x_{i,j}.$$
In this latter case, the regression coefficient $\alpha$ represents the effect of the population size, to the extent that this deviates from a proportionate effect on the response count.  This latter model form also allows you to test the null hypothesis that $\alpha=0$, which corresponds to the population size operating proportionately in the model.
A: I am not clear how you count the "number of people in an treatment facility in a given state/year.", i.e., count the person as one or as 3 if that person went in the facility 3 times in that year. 

If counting as 3, the Poisson regression is good method, and the model can be written as: 
$$Y \sim  \mathrm{Poisson} (\lambda N)$$
$$\log(\lambda) = X\beta$$
where $Y$ is number of people in an treatment facility, $N$ is population size and $X$ contains other covariates. $N$ is called offset in Poisson regression.

If counting as 1, the logistic regression is good method, and the model can be written as: 
$$Y \sim  \mathrm{Binomial} (\pi, N)$$
$$\log\left(\frac\pi{1-\pi}\right) = X\beta$$
The meanings of $Y$, $N$ and $X$ are the same as in Poisson regression.
