Translation:Please do your homework quickly.
Ignore that. I looked it up. Yaru is okay for physical actions (and also can mean give, but usually implies a level of looking down) whereas suru can be used for physical or mental actions. Duo really needs a tips and notes for this, rather than achievements and clubs and whatnot. We can do without those! Priorities, yo.
You can, because "fast' is a "flat/bare adverb" (that is, an adverb that doesn't end in "-ly"). Moreover, the adverb "fast" doesn't even have a competing "-ly" form, so prohibiting its use isn't even an option. However, the traditional view (which came from 18th century grammarians) was that all adverbs should end in "-ly". Quoting Wikipedia, "despite bare adverbs being grammatically correct and widely used by respected authors, it can be advisable to instead use the forms with -ly because these are often considered more socially accepted and of higher status". Very interesting references about this topic are this Grammar Girl's article and this Merriam-Webster's video.
The way I interpret the Japanese sentence is "Please quickly start doing homework" instead of "Please do the honework quickly (e.g. 1 question per min instead of 1 per hour)." The former is 早く and the latter is 速く. So I think the official answer is not a good translation.