The 95% CI of *which* statistic has to not contain zero for that estimate to be significant? Say I want to estimate the difference between the means of two (repeated measures) samples, and the statistical significance of this difference. I have in mind the rule of thumb that an estimate is significantly different from 0 if its 95% CI does not include 0. 
First, am I correct that, with this rule, one does not at all need to compute a p-value?
Second, for the case when the estimate refers to the difference between two means, for it to be significant, the 95% CI of which of the following statistics would not have to contain zero:
1) the difference itself
2) the t-value for a paired t-test 
3) the effect size, e.g. Cohen's d
Third, can it happen that said rule of thumb is incongruent with respect to the three statistics above, e.g. that the 95% CI of the difference between the means does not contain 0 but that for Cohen's does contain 0?
 A: First: calculating a confidence interval (CI) is based on the same underlying premise as the p-value (the desired alpha level, one or two-sided, distribution used to calculate the CI (e.g. normal, t distribution, or proportion CIs)) and should result in the same conclusion when concerning the null-hypothesis as the corresponding p-value. So yes, technically you would not need a p-value next to your confidence interval, but note that this depends on getting the appropriate confidence intervals (and the preferences of certain readers/journals).
Second: in this case I'd say 1) the confidence interval corresponding to the difference itself should not contain 0. Finding 2) (t-test statistic) to be 0 exactly would impossible (as you would need an infinite variance), the same goes for Cohen's d.
Third: as stated under my answer to your first point, the same 1-alpha level confidence interval should correspond to the p-value from a t-test. Thus, those should not differ. AFAIK, cohen's d should also abide by these results. 
