"The dog is biting" would translate to "koira puree", not "koira puraisee". The verb "puraista" (where "puraisee" is conjugated from) is temporally confined, expressing a single bite, whereas "is biting" expresses continuous/ongoing biting, just like "purra" (where "puree" is conjugated from).
The accusative is a bit tricky in Finnish since only personal pronouns as well as the pronoun "kuka" (ken, who) have forms that are clearly accusative (minut, hänet, kenet) - "Näen hänet" (I see him). If you used the word "sipulit" that would just mean "onions" (plural).
The object (accusative) can be expressed in Finnish by using the following cases:
"Syön sipulia." (partitive)
"Syön sipulin." (genitive(like accusative))
"Syö sipuli!" (nominative)
The partitive indicates a partial object, in this case that you do not eat one whole onion, but instead some undefined amount of onion.
The genitive and the nominative indicate whole objects. "Syön sipulin" (I eat the onion) means that you eat the whole onion. The -n ending helps to differentiate between subject and object, since you cannot always tell which one is which by word order alone.
"Minä syön sipulin" (I eat the onion)
"Sipulin syön minä" (I eat the onion - it's me that eats the onion)
"Syön minä sipulin" (I eat the onion)
If you wanted to express that it's the onion that eats the person, you'd then say e.g.:
"Sipuli syö minut" (The onion eats me)
"Minut syö sipuli" (The onion eats me - I'm eaten by an/the onion)
With the imperative ("Syö sipuli!" - Eat the onion!) there is no subject and therefore no need for the -n, and so the object is in the nominative case.