When you say that "he only live[d] with her after his father died", I think this is a misinterpretation of the hierarchical structure of the record. In the earlier records, Olof the son is the crofter, i.e. the "head" of the household, and is listed first, followed by his brother ("co-crofter") and his family, then other relatives (Olof the father and a sister, listed as the father's daughter) and finally other people living in the household. Since they never married, Christina and her children are listed in the last group (Christina is listed as a servant, Piga), and thus are placed close to the father. The first record that was written after the father's death, by coincidence (probably) is also the first record where the son was not the crofter, and as "previous crofter" he is now also listed after the family of the new crofter - and since he has no further family living with him, Christina is listed directly after him (but still as a servant). I see no reason to believe that their relationship shifted due to the death of the father, or that this would indicate that she had had a closer relationship to the father than to the son. (That's not to say that the father hypothesis is necessarily incorrect, of course, but there is no evidence supporting it in preference to the son.)
The note in the right-hand column, as I mentioned in this post, simply talks about needing to hold a separate examination (either for Olof or for his son Pehr Olof), and thus unfortunately does not add anything useful to this discussion.
As for Anders, the birth record says "torparen därstädes Olof Olofsson", i e the crofter in Kojäng. In the HHE record covering 1809, the father is denoted with a "T" for "Torpare", but on the line of the son the word "son" is struck out and the same "T" has been added, and in the next record (1810-1815) it is only the son who carries this distinction, so presumably the father's ought to have been struck out and replaced as well... It doesn't say when they switched, but it seems likely that it had happened by 1 May 1809 when Anders was born. I also note that Anders is listed as foster son in the 1810 HHE record, and my interpretation is that he was fostered by Olof the son (the head of the household) - although it looks like he is grouped with Anna Olofsdotter (sister of Olof the son, and listed as daughter of Olof the father) and her daughter Anna Christina, they have clearly been added after the record was first drawn up, and placed where they are in order to leave space below Anna's brother Bengt and his wife for their children.
So, all in all, I would go with Olof the son being the father.