You can trust SignalTap to a high degree, but it seems to me, that you didn't yet consider the effect of sampling SignalTap input data with the resepctive sampling clock. All described effects are obvious results of this sampling, to my opinion.
To see actual clock waveforms, you have to use multiple oversampling. Your said oscilloscope is sampling the data with 2 to 5 GHz, I guess. A similar time resolution can be never achieved with SignalTap.
There be cases, where SignalTap isn't reliable, when you ignore setup and hold time requirements. But you will get respective warnings in Timing Analysis.