I am trying to estimate the sample size for a national health survey.

From the previous report, the prevalence rates of dental caries and gum diseases are 90% and 85%, respectively. What is the sample size if we are assuming a margin of error of 3% and 95% CI? Also, would the sample size be different if it's a one-sided or two-sided test?

I came across the online calculator; however, I am not sure if they are one-sided or two-sided.

We also have a mean DMFT score of 8.1 with a standard deviation of 7.1 for dental caries. Can I do the calculation by using n =(t X s / E)^2 if I assume the margin of error of 1 unit?

Hi, welcome. You can use a calculator like this one or use the formula. The percentage to estimate ("response distribution" in the link I gave) is usually taken to be 50%. If you're interested in only the percentages you mention, you could use 85% or, conservatively, 80% or something.

Sample sizes are different for one and two-sided tests, but then you'd be in a different framework (proportion testing instead of estimating). Why would you want to do tests? I think estimation should be fine for your purposes.

Your application of the formula for the estimate of the mean is correct.