Este pequeñito = This (very / sweet) little
Grammar:
Demonstrative pronouns used as adjective:
grammar/chapter11_demonstrative_pronouns/11_2_demonstrative_pronouns_adjectival.htm

Diminutive: Diminutive can give nouns and adjectives an emotional turn
grammar/chapter23_aumentativo_diminutivo/23_1_diminutivo.htm

Compró un huevito = bought a (small) egg
Grammar:
Pasado indefinido:
grammar/chapter6_tenses/6_4_formation_indefinido.htm

este = this
Demonstrative pronoun:
grammar/chapter11_demonstrative_pronouns/11_4_demonstrative_pronouns_nouns.htm

lo cocinó = cooked it (the egg)
Grammar:
Pasado indefinido:
grammar/chapter6_tenses/6_4_formation_indefinido.htm

este = this
lo peló = peeled it
Grammar:
Pasado indefinido:
este = this
lo saló = salted it
Grammar:

y este gordito, gordito = and this (very / pretty) little thick one
se lo comió = ate it
Grammar: This is one of the cases, where there is a reflexive pronoun which does not have a semantic meaning. As there is no rule for it, its use depends a lot on the context. We recommend - just ignore it for the time being.

todito, todito = totally
Grammar: Another exception: usually there is no diminutive of indefinite pronouns. Only todo is an exception here.