The "Sig." column in SPSS output for t-test is a two-tailed p-value, i.e. if one want to decide whether to reject a null hypothesis, they need to compare the predetermined significant level with the "Sig." value divided by 2 instead of the value itself.
What about the SPSS output for F-test? It seems that it is a one-tailed p-value (i.e. one can directly compare it with the significance level of a one-tailed test). It would also not make sense if it was two-tailed as F-distribution is not symmetric. But I am still unable to find the confirmation.