Thanks to @nlui, we came up with some ideas that might explain the features that you were seeing in your simulations. Here is a good source to read more about testing convergence to see if the effects that you observe in simulations are physical effects or numerical errors.
I will try to answer your questions in order:
The effective refractive index in the slot waveguide case is a function of wavelength. When you calculate the fundamental TE-mode for slot waveguide, the software picks the central frequency. However, when you propagate the mode and watch the movie monitor, you see the propagation of the light that includes light at different frequencies. For example, you can change the source central frequency and calculate n_eff and you will see some changes (specially if you move from 1um to 2um). This results in a dispersion and a change in the shape of the mode as it propagates along the waveguide. I guess since the strip waveguide is less sensitive to the changes in the wavelength, you are not seeing much of dispersion.
We increased the simulation region and results got improved. The drop in the transmission on the second monitor for slot waveguide seems to be due to numerical errors resulted from the PML. For the strip waveguide, the increase in transmission as a function of wavelength is very small (0.01%). We used a finer mesh and the errors were even decreased, so it is numerical error as well.
By expanding the simulation region, transmission on the first monitor was back to 1 rather than ~ 1.001 in your case.
I hope this could answer your question.