If my normality test is non-significant, am I safe to use the t-test? I took a 30 unit sample from a population. The sample distribution resulted to be normal. Can I state that the population distribution is normal too? If so, with what level of confidence?
 A: We will need to clarify some ideas here. Your sample, being finite, cannot possibly be normal, which is infinite. Also, this quote seems relevant.  

$30$ is a fairly small sample, & the Anderson-Darling test is not the most powerful test of normality to start with.  You may believe that the population is normal as a result, but it certainly isn't proven.  For more information on the underlying topics here, it may help you to read these:  


*

*Is normality testing 'essentially useless'?

*Why do statisticians say a non-significant result means "you can't reject the null" as opposed to accepting the null hypothesis?

Regarding the issue of verifying your assumptions for a $t$-test, what needs to be normal for a paired $t$-test are the differences, not the original data.  You are probably good enough, the $t$-test is pretty robust anyway.  However, the check then test strategy has been criticized.  If you are concerned that the test may not be appropriate, it is generally better to simply use a test that doesn't rely on that assumption, in this case the Wilcoxon.  
