The reaction with p-toluenesulfonyl chloride (TsCl) is known as the Hinsberg test, which is specific for 1$^\circ$ amines.
Only 1$^\circ$ amines (C$_6$H$_5$NH$_2$) react with TsCl to form sulfonamides that are soluble in aqueous NaOH.
2$^\circ$ amines ((C$_6$H$_5$)$_2$NH) react with TsCl to form sulfonamides that are insoluble in NaOH.
3$^\circ$ amines ((C$_6$H$_5$)$_3$N) do not react with TsCl at all.
Thus:
Statement I: False. Not all the given compounds react with TsCl.
Statement II: False. Only the sulfonamide formed by 1$^\circ$ amines is soluble in NaOH.