Some sources (i.e. https://statistics.laerd.com/spss-tutorials/dependent-t-test-using-spss-statistics.php) include assumption about outliers. "There should be no significant outliers in the differences between the two related groups". However, the text book that I read, does not.

My question is, should I check for existence of outliers or normality test is enough to conduct dependent t-test? if yes, what does "significant" outliers mean here?

The number of outliers? Extreme outliers (outlier > IQR3+Q3 / outlier < Q1 - IQR3) ?

Can I still conduct t-test if my data are normally distributed but there are mild outliers? (not extreme)

If my data are not normally distributed I use Wilcoxon signed-rank test. Are mild outliers problem for this test as well?