Who is the biological father for the sons born in cases 5 and 10? Is it some other person (before current marriage) or the person who is marrying her currently?
In order to answer your questions lets First take a look at original Sanskrit shlokas.
स्वयंजात : प्रणीतस्य तत्सम : पुत्रिकासुत: |
पौनर्भवश्च कानीन :
भगिन्यां यश्च जायते ||33 ||
दत्त: क्रीत: कृत्रिमश्च उपगच्छेत्
स्वयम् च य |
सहोढो ज्ञातिरेताश्च हीनयोनिधृतश्च य : || 34 ||
The religious institutes mention six kinds of sons that are heirs and
kinsmen, and six other kinds that are not heirs but kinsmen. I shall
speak of them presently. O Pritha, listen to me. They are: 1st, the
son begotten by one's own self upon his wedded wife; 2nd, the son
begotten upon one's wife by an accomplished person from motives of
kindness; 3rd, the son begotten upon one's wife by a person for
pecuniary consideration; 4th, the son begotten upon the wife after the
husband's death; 5th, the maiden-born son; 6th, the son born of an
unchaste wife; 7th, the son given; 8th, the son bought for a
consideration; 9th, the son self-given; 10th, the son received with a
pregnant bride; 11th, the brother's son; and 12th, the son begotten
upon a wife of lower caste. On failure of offspring of a prior class,
the mother should desire to have offspring of the next class. In times
of distress, men solicit offspring from accomplished younger brothers.
The self-born Manu hath said that men failing to have legitimate
offspring of their own may have offspring begotten upon their wives by
others, for sons confer the highest religious merit.
Who is the biological father for the sons born in cases 5 & 10 ?
Now lets see the definition and meaning of the word Kanin कानीन which is case no 5.
The unmarried girl beget a son by male member of same varna in her fathers house is called as Kanin. So that male person (before current marriage) will be the biological father of that child , and not the one to whom she will be marrying currently or in future. Because the birth of the child has already happned . And the girl is not pregnant if currently she is marrying to some other person.The case will change after her marriage.
In case of 10 the son is called as Sahaod सहोढ
. The girl who was pregnant in kanyavastha ( कन्यवस्था) or while in virgin state , which was latter being given in marriage to the same or different person , while she is still pregnant and delivers a child after marriage with same or different person . So again the person from whom the girl got pregnant will be the biological father of that girl , and not the one whom she is got married later while still pregnant.
3: daughter's son , 6: sister's son
The words used in shloka are Putrika suta ( पुत्रिकासुत) and Bhaginya (भगिन्या ).
Putrika suta - Son of the Daughter . This is a case mentioned in Hinduism scriptures when a person is not having his own sons can accept a son born from his own daughter and the husband of the daughter is pledged or bound to give that son to his Father in law. This is generally finalised before the marriage. In Simple terms a daughter's son who by agreement becomes the son of her father , Such a son is called as Putrika suta . This is mentioned in Manu Smiriti as well as Yajnavalkya smṛti verse Chptaer - 2 -128-132.
Bhaginya - Son of the sister or a nephew is also considered as own son and can inharit property of a person in some special cases as mentioned above , where the person himself is sonless.
So these both meanings you mentioned as per your local language translaton are correct .
This is the Hindi translation
of the two verses by Gita Press . In comment section these terms are discussed. Page 356.