It sounds like you are intending to test the hypothesis, "Does skill level on the test increase with increasing educational attainment?" This corresponds to the null hypothesis: avg skill of ed. level 1 = avg skill of ed. level 2 = avg skill of ed. level 3 = avg skill of ed. level 4 = avg skill of ed. level 5.
To answer your first question, a 1-way ANOVA will test whether there is a difference amongst these groups, but will not capture the ordinal nature of the educational attainment. An alternative approach would be to treat educational attainment as continuous and use a linear regression.
For your second question, let me clarify: you have not yet gathered data, correct? You are asking how many participants you need in your experiment. For this you need to conduct a power analysis. For a linear regression that is a little tricky to do, but for a 1-way ANOVA it's pretty easy. If you do not have access to statistical software there are a number of online calculators. One I found quickly is: https://www.anzmtg.org/stats/PowerCalculator/PowerANOVA
Your third question is just a variation on the second question. When you do the power analysis you will find how many subjects you need for your desired power level.
To do the power analysis you will need to specify the standard deviation you expect and the difference in mean score which you are testing for. The difference in mean score is easier because you should be able to know what type of difference in your scale is meaningful. If you don't know the variation, it would be a good idea for you to try out the scale on some test subjects first (a mix of educational levels) and calculate the mean and standard deviation. You can use this for the power calculation. Alternatively you can just use an effect size in the calculation, meaning that for a strong effect (effect of 1.0 or larger) you need the change in mean to be proportional to the standard deviation. A moderate effect is 0.5, meaning the change in mean is half the standard deviation. Effect sizes of <0.5 are usually considered small.