I have been reading about secret sharing schemes, and they regularly come up with a line that says 'that a person upon receiving a state measures the state in one of the basis say the computational basis $|0\rangle,|1\rangle$ or the Hadamard basis $\dfrac{|0\rangle+|1\rangle}{\sqrt{2}}$ and $\dfrac{|0\rangle-|1\rangle}{\sqrt{2}}$. Now what I don't understand is suppose I have state say $$\alpha|0\rangle+\beta|1\rangle$$ that I get from someone and that someone will give me the secret when I 'measure' the state in the right Basis.

1. Now If I chose the standard basis $|0\rangle,|1\rangle$ what will be the result I will get? If I measure with respect of the projection operator $\langle 0|$ I get $\alpha$ and If I measure with respect of the projection operator $\langle 1|$ I get $\beta$.
2. If I measure with respect to the Hadamard basis $\dfrac{|0\rangle+|1\rangle}{\sqrt{2}}$ I get $\dfrac{\alpha+\beta}{\sqrt{2}}$
and if i measure with respect $\dfrac{|0\rangle-|1\rangle}{\sqrt{2}}$ I get $\dfrac{\alpha-\beta}{\sqrt{2}}$.
3. My first question basically means just selecting a Basis wouldn't serve the cause, because the basis themselves involve different projection operators and hence different operators. can somebody explain this concept?
4. Suppose I have a state $\omega|0\rangle$ and I measure in the standard basis then I either get a $0$ or $1$ depending on the projection operator chosen. So does that mean that $\omega$ has nothing to do with the result?