A1: it is not neccessary to do convergence for every design. Only do this for the promissing design.
A2: you have to open the analysis group. open the analysis script, and run it. The error will be displayed so you will know what is the cause. Most likely one or more time monitors has no data, out side of the simulation region. This can be true when you use symmetry BCs.
A3: override mesh is mostly used for resolving geometry details. I would suggest to have about 4~6 for moderate accuracy, 2~3 for low accuracy.
Since the photonics crystal is supposed to be periodic, so you will need to make sure the lattice is discretely periodic, which can lead to better result. eg, you will see each hole is meshed the same.
A4: if the simulation dimension changes the result, it means the original dimension is too small. You will need to make sure the PML is about half-wavelength away from the structure.
It may also mean the Q analysis group or dipole cloude is not properly set. Actually such simulation will need iteration simulation: find the initial resonance wavelength, then add a monitor to monitor the field distribution for that wavelength. then move the dipole and time monitor to be in high intensity spots. This can give you a better result. You will need to repeat this.
Make sure the silicon is embeded in a dielectric material, other than air, for practical reason.
A5: sweep is to find the range of a parameter. I suggest to not sweep many points. Instead, iterating the parameter ranges to find the range of global optimization. then you can use optimization to get best result.
Please note that the forum is for one-post one solution. So please post your most difficult quesiton once a time in one post. Once it is solved, post another question.