1) Does it make sense to use a fixed effects model on cross sectional data (note, not time-series cross sectional data) if the goal is to account for observed and unobserved heterogeneity among clusters?

2) Would the resulting coefficients have the same interpretation as in the usual fixed effects context (where there are repeated observations on the same level 1 units over time)? (see example below)

A concrete example would be data that consists of individual subjects sampled from different countries. The samples are spread across multiple time points, but each individual is only observed in one time point. So, we could have 10 countries, with 90 individuals in each country, and three time points with 30 individuals in each. Then, we could fit a model like:

y_ij ~ Gaussian(mu_ij, sigma)
mu_ij = beta_0 + beta_1:9 country_i + beta_10:11 time_ij

where country_i consists of 9 dummies (betas 1 through 9) for our 10 countries and time_ij consists of 2 dummies (betas 10 and 11) for our 3 time points (that are within each country).

Relating to question #2: how would betas 10 and 11 be interpreted here? I know they'd be average differences in the response between the reference time point and the other 2 time points. But would this be for a 'typical' country? Averaged across countries? Within each country? (and if this last one, what does 'within each country' actually mean?)