This is an interesting question. Indeed, understanding random effects $u_i$ as draws from a larger population is well in line with the underlying model, which treats them as a random variable, e.g.:

$u_i \sim_{iid} Normal(0, \sigma^2)$

But there are arguments for treating entities as coming from a larger population even if your sample is a full census. Any realized set of entities can be viewed as "chosen at random by Nature", an argument made by Freedman (2005). Relatedly, Deming (1953) has argued that if a full census is used to solve a, what he calls, analytical problem, i.e., when inferring an underlying more general relationship or process with the goal to generalize (as in your case), even a full census should be treated as a sample with sampling error. This would also justify the view of your counties coming from a larger distribution (in Deming's words, coming from the "causal system" that produced it).

It will probably depend on your discipline how strictly people treat the different assumptions made in random effect models, but the random effect independence assumption is probably more relevant and empirically verifiable than what you view as your population of counties. Given the efficiency advantage of RE over FE models and the convenient estimation of between-county variance in RE models, which is one of your side goals, I would advise for the RE model, provided you can credibly argue that the independence assumption is sensible in your context and does no harm.

References:

Freedman D. A. (2005). Statistical Models: Theory and Practice. Cambridge, UK: Cambridge University Press.

Deming, W. E. (1953). On the distinction between enumerative and analytic surveys. Journal of the American Statistical Association, 48(262), 244-255.

This is an interesting question. Indeed, understanding random effects $u_i$ as draws from a larger population is well in line with the underlying model, which treats them as a random variable, e.g.:

$u_i \sim_{iid} Normal(0, \sigma^2)$

But there are arguments for treating entities as coming from a larger population even if your sample is a full census. Any realized set of entities can be viewed as "chosen at random by Nature", an argument made by Freedman (2005). Relatedly, Deming (1953) has argued that if a full census is used to solve a, what he calls, analytical problem, i.e., when inferring an underlying relationship or process with the goal to generalize (as in your case) rather than just count, even a full census should be treated as a sample with sampling error. This would also justify the view of your counties coming from a larger distribution (in Deming's words, coming from the "causal system" that produced it).

It will probably depend on your discipline how strictly people treat the different assumptions made in random effect models, but the random effect independence assumption is probably more relevant and empirically verifiable than what you view as your population of counties. Given the efficiency advantage of RE over FE models and the convenient estimation of between-county variance in RE models, which is one of your side goals, I would advise for the RE model (given other assumptions, like the independence assumption, make sense in your context and do no harm).

References:

Freedman D. A. (2005). Statistical Models: Theory and Practice. Cambridge, UK: Cambridge University Press.

Deming, W. E. (1953). On the distinction between enumerative and analytic surveys. Journal of the American Statistical Association, 48(262), 244-255.