Identifying August Edvard's father may be near impossible when there is no explicit mention in the birth records, but there could be other clues. Note that he was born in Ramsberg parish, and that Carolina formally moves to Ramsberg only three years later (to marry Lars Erik Larsson, who was already resident in Ramsberg). I haven't checked where August Edvard's siblings were born or whether they provide clues as to who was the father of each, but I'm of course hinting at the possibility that Lars Erik Larsson may have been father to all three of them, before he married their mother.
I recall that the clerical survey record listed them as born f.ä (före äktenskapet, before marriage), while other children may be listed as oä (oäkta, illegitimate). There may be an actual distinction here, or it may be just a random variation in the records, I'm not sure. Note also that when August Edvard emigrates in 1888, the emigration record states his last name as Larsson, suggesting that Lars Erik Larsson is indeed his real father, or that he was at least adopted as the son of Lars Erik. I haven't checked what last names the other siblings used. We may never know the real reason, unless there is an explicit note about it in some record (an estate inventory, or whatever).
Investigating this may take time, and since you will be visiting Sweden in only four weeks, I suggest putting the question of August Edvard's biological father aside for the moment, and concentrate on the part of the family we now know he had, and the places where he lived, so that you can get the most out of your visit.
Yes, I assume the 1924 death records for Ramsberg are available as well, but they usually don't contain a lot of information, just the name, age and residence of the deceased, date of death, cause of death, date of burial, and the name of the priest who performed the burial. If you are lucky, there may be a family grave in Ramsberg, but it may require some detective work to find it.
By registered residents I'm referring to those listed in the clerical survey rolls (husförhörslängd) or parish registers (församlingsbok), depending on whether we are talking about the 1800's or the 1900's. They are the ones I used above (labelled AI and AII respectively). In the old days, the church was supposed to know exactly who were residents of each parish, and individuals who moved from one parish to another had to obtain written proof of their identities (flyttningsbetyg). People were still allowed to travel or stay for extended periods of time outside their home parish, but when children were born outside their mother's home parish (or people died outside their own), this lead to written communication between the parishes to list the child in the proper book.