"Five of the Solomon Islands disappeared" due to sea level rise, how is this possible so quickly?
The sea is not the only thing that rises, the sea floor can also rise and fall in accordance with the underlying geology. Oceanic tectonic plates sink as they age (and thus get colder and denser), the Solomon islands happen to be on a  sinking section of the ocean floor and at the same time the sea level is rising. Keep in mind many of those islands have a very shallow slope, so a few rising millimeters are many feet of inland horizontal motion. Of course once the wave base reaches  new "virgin" sediment (often loose sand), it often washes it away moving the tide line even further inland. 
Storms cause a storm surge which can wash away sediment. A rising sea level means the storm surge is reaching sediments they could not reach before, so they are causing faster erosion. In addition, salt water kills a lot of plants further eroding the sediment, so a higher surge means more dead plants and more settling of sediment. Storms can cause a tremendous amount of erosion on unprotected soil, a single storm washed away two entire city blocks in the Atlantic City history. 
I am not entirely sure, but prevailing winds can cause some displacement of water (the ocean is not completely leveled because it is moving) so a change in the trade winds could lead to a minor effect of local sea level, how much this contributes, I have no idea.  


Note: the OP started with a numbered list and specifically asked about the highlighted numbers, hence only 2-4 were referenced in my answer (the first one was a normal sea level rise).