So I take it since the hover-hints also suggest "him" and "her" (ona) that this serves not only for the nominative (mianownik?) but also for either the accusative (biernik?) or dative (celownik?). Is that dual-use correct, that on/ona are NOT only used in the nominative (like he/she are only used that way in English and have him/her for the others?)
Mianownik/Nom. (kto? co?) - on/ona (subject of the sentence) - he/she Dopełniacz/Gen. (kogo? czego?) - go/jego (short/long form)/jej - his/her Celownik/Dat. (komu? czemu?) - mu/jemu (short/long form)/jej - him/her Biernik/Acc. (kogo? co?) - go/jego (short/long form)/ją - him/her
Him = Jemu
I gave him an apple = Dałam/dałem jemu habłko
Her = Jej
I gave her an apple = Dałam jej jabłko.
On & jemu are not in the same tence.
You can ask "who? what?" (kto? co?) for nominative and the answer will be "on". And there are questions for every tence (search for it, it is very helpful).
Wikipedia might help you a little. Also, don't worry, cause sometimes even Poles have problems with tences :) (https://en.m.wikipedia.org/wiki/Polish_grammar)
him - mu (short form)/jemu (long form)
The sentence "Dałam/Dałem jemu jabłko" is not
correct here, because it requires the short form:
I gave him an apple - Dałem/Dałam mu jabłko
her - jej (there is one form only)
I gave her an apple - Dałem/Dałam jej jabłko
"on & jemu are not in the same case"
"Many Polish native speakers use wrong cases" :-)
"Ono" refers to neuter gender in Polish grammar, so
you can say:
"Ono/Dziecko płacze" - He/She/A/The baby is crying
Note: English "it" refers to animals and objects, so both sentences "To płacze" and "It is crying" sound very rude,
or inappropriate when referring to a human child.