I don't understand why we have to write the key signature on the bass line? It's already precise on the treble line, so we already know the tonality of the piece.
What's more, is it possible in this case that the accidentals of the treble and bass clef differs for the same piece?
Like many things in music, the answer is "because that's the way everybody does it, and if you didn't do it, too, your music would look strange and confuse people." To your second question, there are a few pieces that have a different key signature for the top and bottom staff. Parts of Bartok's Mikrokosmos, for instance. Thus, if you didn't put the key signature on both, people might think that it applied to only the top staff.
What if the music uses transposing instruments?
What if the music is polytonal?
You're taking your limited view of piano grand staff, treating it like a single stave, instead of two "players", your right and left hand, and then imagining a notation scheme that won't work for a lot of instrument combinations.
Just to be clear that we aren't talking about something that applies only to unusual scores, here is an example from a set of minuets by Mozart, K. 568...
...where three key signatures are used, zero sharps/flats, one flat, and three flats. Three key signatures are used, because the wind instruments transpose. Sometimes a score can have key signatures using both sharps for some instruments and flats for others, depending on the instruments. Here is an example from Brahms, Symphony No. 2...
...it's in D major, but notice the three key signatures: two sharps, one flat, and zero sharps/flats.
The basic idea is each staff represents an instrumental part, and those parts must be given their key signatures, which may differ from other instruments.
Think of your piano grand staff as two instruments, your left and right hands, but because those two "instruments" don't transpose, they will have the same key signature.
The fact that your key signature sharp/flat count doesn't differ between staves for piano, doesn't mean that's how it works for other instruments.
You have to put the key signature on every stave, because no sharps or flats is a key signature -- C maj.
You would have to invent a convention that indicated "Refer to top stave for key signature"
Well, I would actually challenge you on the conclusion of this statement:
It's already precise on the treble line, so we already know the tonality of the piece.
Key signatures don't necessarily imply the tonality of the piece directly.
To give an example, suppose you have a piece with a key signature of 0 sharps and 0 flats. This could imply the key of C major. It could also just as well imply A minor. These keys and tonalities are not the same thing. Yet, they both have the same key signature.
What the key signature actually tells you is which notes are sharp and which ones are flat in the piece. Most people use the standard set of key signatures, but technically you don't have to use the standard set of key signatures. While most composers probably should use the standard, there are a few examples out there where they don't. It's not entirely unheard of.
Thus the notation is open to have something like different key signatures for each hand. So by putting the same key signature on both, you are implying that something weird isn't happening.
That said, my guess here is that one of the main reasons (and a rally good one in my opinion) why we put them on both is for readability. If the right hand has one key signature, and the left has none, but this actually implied the key signature for both it would be harder to read. I would much rather have the key signature on both instead of having to remember the key signature applied to both hands. This would be especially true in sight reading situations.
TLDR 1) Technically speaking, it's not truely redundant and 2) It's easier to read and remember if it's on both hands.
Your second question mentions accidentals and, although this is different to key signatures, it is a valid question.
The simplest case is; in a minor key, with the melody in the right hand, and the left hand accompanying with the harmony. The harmony on the lower stave may have one accidental to sharpen the seventh. The melody on the top stave may have two accidentals, sharpening the sixth and the seventh. Thus the two staves might have different accidentals.
The accidentals are not necessarily applied to every occurrence of these notes, but even if they are, they are not usually added to the key signature.
In (slightly) more complex pieces, the melody can stray onto the lower stave, and harmonic lines can be found on the upper.
Of course, accidentals can be used for other purposes too.
