I also wondered if "putus," nearer the top of this definition, is related to the Spanish "puta" and, if so, why?

No, Fr putain, Sp puta 'whore', and It putto are from Latin puttus 'child'. Meyer-Luebke suggests that the semantics went 'child' to 'bastard, Schandekind' to 'whore'. There's a Latin verb puto, -are 'to cut' which may or may not be related to puto 'to clean, cleanse, purify' which is related to purus. The sense for puto 'to cut' coming from a gardening usage of 'to weed'. So, Pokorny gives three separate roots in PIE: (1) *peu- : *pewe- : *pu(:)- 'to cleanse, purify; refine, clarify; sift, strain' - whence Skt punati 'to purify, cleanse', Latin purus 'clean, pure', purgo 'to make clean or pure, purify', and Irish ur 'fresh; noble'; (2) *pe:u- : *p@u- : *pu(:)- 'to hit; to sharpen'; (3) *po:o- : *p@u- : *pu(:)- 'little, slight, few' perhaps also 'youth, animal young, small animal' *po:(u)-lo-s 'youth, boy' *p@u-ko- 'few'. *pu-tlo-s 'child' : whence Skt putra 'boy', Greek pais, paidis 'child', Latin puer 'boy', puella 'girl', puttus 'child'. Because these roots all look similar, I guess they were posited on semantics.