1. Complex numbers are especially useful to describe systems dealing with rotation, or if you want to describe a 2-dimensional space, corresponding to a single variable. Instead of x/y, you can just have a real/imaginary component to x. The entire qubit state space (Bloch sphere is a great visual) is a 3 dimensional sphere with a radius of 1, and because that space will output a binary output that splits the sphere in half along some plane, 2 2-D vectors can describe the entire space, with a 2-D vector corresponding to each measurement output possibility. That being said, a 3-D vector could describe the entire system instead (though this would not be capable of describing global phase), but I believe that the common dual vector system is primarily favored due to the nature of the binary outputs. Each 2-D vector has a norm (amplitude) related to the probability of that measurement, and some single 3-D vector system would require more math to get those amplitudes. Alternatively, the real and imaginary components of each 2-D vector could be described individually as 4 real values (this is overkill, since we only need 3 real values to describe the whole system, unless we are considering global phase), but does that improve anything? Consideration would still need to be made to respect norm along each vector, and the values would correspond to directions orthogonal to each other. This effectively describes a complex number.
2. A classical system of n bits also has 2^n _possible_ states, and it "occupies" one of those states at any time, based on the concrete value of the classical bits. A quantum system of n qubits has 2^n _possible_ states, and it occupies X states (where 1 <= X <= 2^n). All n qubits could be set to pure states (say |000....0>), and this would mean that the entire system occupies one single state within the 2^n space. Alternatively, every single qubit could be set to a superposition (like |+++...+>), and this means that the system is "occupying" all 2^n states via superposition _simultaneously_. Now, of course, measuring such a system in the 0 1 basis will simply give you a random state from the list of 2^n possibilities. But, _before_ you measure, there are amplitudes corresponding to 2^n states. Manipulating those amplitudes for some computational goal is quantum computing.
A metaphor I've used to help myself understand superposition is flipping a coin. Before flipping, there are two possible outcomes, and I could say that an unflipped coin is "both" Heads and Tails. Once the coin is flipped (even if you haven't looked yet), the coin is either H or T (and _not_ both). Similarly, two unflipped coins have 4 potential outcomes, while 2 flipped coins is again a single state (one of: HH, HT, TH, TT). The space of possible outcomes remains the same, but the flipped coins are only a single state in the space. The key is that before measuring (flipping the coin), the only indication that a qubit has of its future value is stored in the qubit's amplitudes. This is why quantum speedups rely on manipulating interactions between amplitudes, like amplitude amplification used in Grover's Algorithm, or the Quantum Fourier Transform. If this metaphorical explanation isn't terribly useful, I'd encourage you to read others' descriptions of superposition. Unfortunately, I haven't found a great amount of support for describing superposition in the English language.
Alternatively, the dimensions of a space are defined as the number of vectors in that space that are all orthogonal to each other simultaneously. One great way to prove orthogonality is the Pythagorean theorem. For a single qubit system, we know that the amplitudes (a and b for simplicity) are related to the probability via a^2 = (P of a), and b^2 = (P of b). Because the measurement outcome must be either a or b, we can also say that a^2 + b^2 = 1. This generalizes to multiqubit systems, where summing the squares of each measurement outcome amplitude will equal 1. This implies that a superposed 2 qubit system (|++>) is described as a 4 dimensional vector in a 4 dimensional space, and not just a 1-D vector within that same space. A classical system has no equivalent multidimensional vector system, just concrete values (corresponding to a 1-D vector pointing along some axis in this visualization).
2. A classical system of n bits also has 2^n _possible_ states, and it "occupies" one of those states at any time, based on the concrete value of the classical bits. A quantum system of n qubits has 2^n _possible_ states, and it occupies X states (where 1 <= X <= 2^n). All n qubits could be set to pure states (say |000....0>), and this would mean that the entire system occupies one single state within the 2^n space. Alternatively, every single qubit could be set to a superposition (like |+++...+>), and this means that the system is "occupying" all 2^n states via superposition _simultaneously_. Now, of course, measuring such a system in the 0 1 basis will simply give you a random state from the list of 2^n possibilities. But, _before_ you measure, there are amplitudes corresponding to 2^n states. Manipulating those amplitudes for some computational goal is quantum computing.
A metaphor I've used to help myself understand superposition is flipping a coin. Before flipping, there are two possible outcomes, and I could say that an unflipped coin is "both" Heads and Tails. Once the coin is flipped (even if you haven't looked yet), the coin is either H or T (and _not_ both). Similarly, two unflipped coins have 4 potential outcomes, while 2 flipped coins is again a single state (one of: HH, HT, TH, TT). The space of possible outcomes remains the same, but the flipped coins are only a single state in the space. The key is that before measuring (flipping the coin), the only indication that a qubit has of its future value is stored in the qubit's amplitudes. This is why quantum speedups rely on manipulating interactions between amplitudes, like amplitude amplification used in Grover's Algorithm, or the Quantum Fourier Transform. If this metaphorical explanation isn't terribly useful, I'd encourage you to read others' descriptions of superposition. Unfortunately, I haven't found a great amount of support for describing superposition in the English language.
Alternatively, the dimensions of a space are defined as the number of vectors in that space that are all orthogonal to each other simultaneously. One great way to prove orthogonality is the Pythagorean theorem. For a single qubit system, we know that the amplitudes (a and b for simplicity) are related to the probability via a^2 = (P of a), and b^2 = (P of b). Because the measurement outcome must be either a or b, we can also say that a^2 + b^2 = 1. This generalizes to multiqubit systems, where summing the squares of each measurement outcome amplitude will equal 1. This implies that a superposed 2 qubit system (|++>) is described as a 4 dimensional vector in a 4 dimensional space, and not just a 1-D vector within that same space. A classical system has no equivalent multidimensional vector system, just concrete values (corresponding to a 1-D vector pointing along some axis in this visualization).