The only way that I know of to do this, requires spatial Analyst so perhaps I'll write it up as such and if it is possible another way, someone else will write that up as a separate answer. With the Spatial Analyst Extension, the tool to use is Zonal Statistics. I typically use Zonal Statistics As Table.
I typically use a projected coordinate system whenever I do anything of this nature especially if area is important. It may not be entirely necessary to re-project the data to do this (you can certainly try). If it doesn't run well in GCS, then try to re-project and see if that fixes the problem.
The method will count cells within the polygon that includes most of the cell (to answer your question in item 1.
Use the Zonal Statistics As Table Tool with the polygon layer as your zone (select a field that uniquely identifies each polygon as the zone field). The raster is your value layer.
The results will have values for each different category. You have the option of grouping values in the raster first or in the table after (whichever you find easiest). You can use a tool like "Reclass" to reclassify the raster into groups (like your example of >50) or you can certainly do it with the table after running it on the original raster.
ADDED: To address your comment about how zonal stats works and spatial resolution, a good practice is to set your Environment Settings (Raster Analysis Settings specifically). Set the cell size and snap raster to your input!
Also to answer your question about boundaries, it converts the polygon to raster first so setting the snap environment helps. The raster will never conform to the shape of the polygon but it will assign values to the zone raster based on the values of the cells using the zone that contains the "majority" of a cell.
Since your raster is what it is and your polygon is what it is, this is really the best you can do. The higher the resolution the raster is, the better it will work.
SUM should work as a statistic for you AFTER you run CON or reclass values >50 to 1.
Hope this helps.