Bredvid is 'next to' like in 'side by side'. vid is in the general vicinity of something.
bredvid makes the objects more equal, they're next to each other.
vid is an unequal relationship: one thing is seen as a landmark for another thing. This makes it unnatural to use in some combinations: if the thing that something is near is smaller or not stationary, vid is not a good match.
near is nära in Swedish. It expresses closeness but not the side-by-side feeling in bredvid.
near and nära are more general so they don't necessarily imply the asymmetric relationship that vid does.
As long as an animal has feet (well, or something like it: paws, hooves, whatever) and is in a standing position, it's normal to say that it stands in either language.
Because, in both English and Swedish, the pronunciation of words naturally drifts and drifts. People just don't say words the same as the previous century. However, the spelling doesn't get updated very often. People like Webster come along and try to bring spelling reform, but that makes all the old printed texts outdated and it also makes standard a drift that some speakers may never have adopted.
I might add that the last time Swedish had a spelling reform was in 1906, when the spelling of the sounds T and V where simplified to T and V (rather than dt, hv, f, fv etc)
Some silent letters were kept though, especially before word-initial J, which is why we have words like djur, hjärta and ljus where the initial letter is silent.
Also, many short E sounds were changed to be spelled with Ä. (elg, enka, ega -> älg, änka, äga.)
I think I can see how that might be grammatical with stood as a past participle. With is stood being similar to is placed. "The sign is placed in the middle of the field". In cases like this, the sign is an object, not a subject. Someone acted on the sign. It did not come to life and stand itself in the field. For this reason, I would not say the dog "is stood" next to the horse, unless it is a statue and someone placed it there.
In this Swedish exercise, the dog is the subject, not the object, and so I would call it a bad translation to make it an object in the answer.