Cut me, I am still normal.
Change my colour, I am still normal.
The only way you can make me change is to draw on me.
Change one of the letters and I am a dog.
Change another one and I am poor.
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Paper
Cut me, I am still normal.
cut a piece of paper and you just have two pieces of paper
Change my colour, I am still normal.
Colored paper is still paper
The only way you can make me change is to draw on me. I don't really have an answer for this, but just a thought
If you draw on a piece of paper - it's no longer just paper - it can be considered a piece of work/art
Change one of the letters and I am a dog.
Piper - the aiport K9 dog? link
Change another one and I am poor.
Pauper - this is technically adding a letter, so may not be it
Are you a
DOT?
Cut me, I am still normal.
Change my colour, I am still normal.
The only way you can make me change is to draw on me.
Mathematically, a dot is a point with zero dimensions, zero distance in any direction. So you can cut it, but half of zero is still zero, it's the same thing. You can change its colour, but that only needs a zero amount of ink, so it's still the same thing. If you draw on it, though, then you can extend it to a line or a curve, which will be something different.
Change one of the letters and I am a dog.
DOG is one letter different from DOT.
Change another one and I am poor.
SOT is one letter different from DOT. A sot might be poor, having wasted all his money on alcohol?