I am looking for sources for the prohibitions of squeezing a sponge on shabbat and also grinding salt or pepper. Are these “deorayta”, and if so which of the avot melachot do they relate to?
Squeezing a sponge, if one wishes to collect its water, is a rabbinic prohibition, which is derived from the Torah prohition of dash.
The prohibition of dash involves ‘revealing’ the grain that is hidden within the chaff and shell in which it grows, and from here the Sages extract that the procedure of ‘revealing’ trapped liquids, such as ‘revealing’ the wine trapped inside grapes or the oil trapped inside olives, is a Torah prohibition. It is therefore a Torah prohibition to squeeze grapes and olives (for other fruit, the prohibition is only rabbinic, because the juice is not considered to be trapped inside the fruit, but it is rather created by means of pressing the friut), or to milk a cow.
For a sponge, the prohibition is only rabbinic, because the sponge did not come into being with water trapped inside it, but only absorbed the water later. However, there could also be a Torah prohibition involved, not of dash, but of libun, cleaning. This is because squeezing out the water is a stage in cleaning the dirt off the sponge.
Grinding salt and pepper is a Torah prohibition of tochen, grinding. Although some maintain that there is no Torah prohibition for salt, because is does not grow from the earth, the concensus of most authorities is that there is a Torah prohibition for both salt and pepper, provided they are ground in a normal way.
Sources: Concerning grinding, see Shabbos 141; Shulchan Aruch, Orach Chaim 321:7; Tehillah Ledavid 7; Misnah Berurah; concerning squeezing, see Shabbos 143 and commentaries.