Just a quick recap after 18.104.22.168:
Collecting the number of unique raster values within a specified radius mask: TileFilterDiversity(...)
Collecting the difference between maximum and minimum raster values within a specified radius mask: TileFilterMax(...) - TileFilterMin(...)
Clump and sieve: no functions that output connected / cleaned up regions as a raster, but you can do TileTraceArea to extract all areas with different values, then you can remove areas that are smaller than a specific threshold. There is no way to convert the areas back to raster for now - we do not yet expose that as a function, but we might in the future. We might also add functions to do clump / sieve on rasters directly without dropping into vectors.
Any links for opening / closing (I am afraid of mixing things up)? Is this something like - "given value X, if a pixel not equal to X has X in one of its neighbors - however these neighbors are selected - set the pixel to X"? If so, we don't have that yet, but that's easy to add and we were planning to add something similar. If you could point us to what you are after specifically, that would help.
Added later: noticed the mention of erosion and dilation near opening and closing - all is clear now. We don't have this yet, but we will have it. In the meantime one can (fairly easily) do these operations using a script function that takes a tile, performs the operation, returns the result back as a tile, then call that in a query.