A p-value is the probability of seeing a result at least as extreme as yours if the null hypothesis were true. Enter a z-score directly, or an observed value with the mean and standard deviation, and choose a one- or two-tailed test. The result comes with the tail area it represents and what it does and does not let you conclude.
If the null hypothesis were true, there is a 3% chance of observing data at least this extreme. It is not the probability that the null hypothesis is true, and it is not the probability that your finding is a fluke.
Use two-tailed unless you have committed in advance to testing a direction and genuinely do not care about an effect the other way. Switching to one-tailed after seeing the data halves the p-value for no good reason and is a well-known way to manufacture significance.
No. 0.05 is a convention, not a law of nature, and a p-value says nothing about effect size or practical importance. Report the effect and a confidence interval alongside the p-value.