Here I think it's just a case of an intransitive verb following the rules and taking "Det" as an object as there is no direct object. [I do not agree (to it)]
The full translations are shown in the cases of specific usage (for eg: "Hvordan går det?" or "kan godt lide") where, as you've said, we cannot guess.
Thank you for asking. In another exercise we have the sentence "Barnet synes ikke om maden" which was translated to "The child does not like the food". It seems that the "på" makes the difference. So your sentence should translate to "Jeg synes ikke om det". But I am not sure. Edit: it is "synes om", not "synes på".
I got this sentence first as a 'write what you hear' but I don't think I would've been able to translate that to English because I wouldn't have thought to use the word "so". No words in the sentence translate to "so", so why would I put it there? I would read this sentence as "I do not think it" as those are the words in the sentence. It's just another one of those quirks of languages that are the reason I'm terrible at learning them, I guess.
Tror is to think or have an opinion about something without any direct observational evidence for it. For example, if you and a friend are looking at a shirt and debating whether or not it is blue or purple, you would say JEG SYNES SKJØRTEN ER LILLA. But, if you have not seen it and you are basically just guessing, then you would say JEG TROR AT SKJØRTEN ER LILLA. I think. I am not a native speaker. If I am incorrect, hopefully someone will step in and correct me.