I'd say all are true.

Regarding a) You can increase the spatial resolution of an image by resampling. Of course, this doesn't automatically increase the image quality. If you want to get higher detail in a multispectral image, you can perform pansharpening or image fusion with a higher-resolution scene (first link).

Regarding b) Smaller variations in image acquisitions could theoretically lead to different spatial resolutions of images from the same sensor. But in most of the cases, images are provided as products which have already been resampled to a defined resolution (10/20/60 m for Sentinel-2, 15/30 m for Landsat ect.). If you however work with radar (SAR) images, pixel sizes can vary decesively according to the incidence angle of the acquisition. Again, this only addresses SLC products, while GRD products are already resampled to a defined resolution.

Regarding c) If you process imagery with SNAP, ERDAS, ENVI ect. many of the tools require recalculation of the pixels. Sometimes this also includes new pixel geometries. But all of the mentioned software offer to define the output resolution of each processing step so spatial resolution won't change if you don't want that.

Generally, when speaking of spatial resolutions and pixel values, resampling is an important thing to keep in mind because it defines how pixel values are calculated in the case of changing spatial resolution (second link).

http://www.geosage.com/highview/imagefusion.html

http://web.pdx.edu/~jduh/courses/Archive/geog481w07/Students/Craver_Resampling.pdf