Hi Luca,
Here is the structure:
【S.+(几乎)快+V.+完+O.+了。】OR【 S.+O.+(几乎)快+V.+完了。】
我快写完作业了。or 我的作业快写完了。
他快做完饭了。or 他的饭快做完了。
你快考完期末考了。or 你的期末考快考完了。
here to answer your questions,
So which sentences are correct?
1- 我快写完作业了
And what about verb object?
3- 我快做”完“饭了 or 我快“煮”完饭了
4- 我几乎做到饭 (This sentence is incorrect.)
Can i use 完成 in any way?
No, you can not use 完成 in this case. Although it means "complete" in English, it usually means something's about to be formed completely. For example, 我的作品快完成了。My (art) work is almost completed.