As you already know, a census implies that you get information about every single unit in your population. Since you will have full information on your subjects, there is no reason to perform estimations: what you get is what it is. I'll try to answer your questions based on that principle, I hope it is useful
1) Missing data can become problematic, particularly since it will alter your calculations and the number you get won't be the real value of your parameters. Missing data will not turn your census into a sample, it will directly distort the results. Procedures for handling missing data, such as imputations should be appropriate since, depending on the mechanism of missing data, your information could be biased.
2) Variance is relevant as a measure of dispersion. You can still calculate variance and standard deviation and interpret them accordingly. Estimation is irrelevant, since the standard deviation you get is the real standard deviation, so you wouldn't be using an estimator, but you should calculate the parameter directly. You don't have to look for significant differences then, if you get 10 one year and 11 on another, those are different, period. As you mentioned, there is no sampling error in a census, but there are many non-sampling errors that may occur (problems with surveyors, poorly designed questionnaires, etc.) and that can alter your results.
3) I don't really get this question. As I told you, statistical significance is a concept used in samples, I don't think it would be most useful in a census.
4) A census can only occur in a finite population. As you know, a finite population is well defined and every member can be appropriately identified. An infinite population means you don't know the size or the actual members of your population, so you can't adequately produce a census.