If electrons have a negative spin of 1/2 do all particles also have the ability to have a negative spin?

In quantum mechanics, spin is a kind of angular momentum that is intrinsic to a particle. Like normal angular momentum, when we go to the quantum world there are two numbers that describe the spin of a body, which we label $s$ and $m_s$. $s$ can be integers or half integers, depending on what you're trying to describe, and $m_s$ is an integer or half integer that takes on values from $-s$ to $s$, differing by at least $1$.

The spin value quoted in these particle lists you are looking at is the $s$ value, which gives you the magnitude of a particle's spin vector through the relationship $\sqrt{s(s+1)}$, which can be derived from quantum mechanics. This is the value always quoted for the spin of a particle.

The $+\frac{1}{2}$ and $-\frac{1}{2}$ are the values of $m_s$ and they are the components of the spin along some axis, usually the z-axis.

This is a case where our use of the word "spin" is a little sloppy. We use it to describe both the spin quantum number which is 1/2 for electrons, and also to describe the component of the spin vector along the z-axis which for electrons can be either +1/2 or -1/2. It is usually obvious by the context which of these two usages is meant, but it can be confusing for a beginner. In one case we are referring to the eigenvalue of the spin operator squared ($S^2|>=s(s+1)|>$) and in the other case to the eigenvalue of the z-component of the spin operator ($S_z|>=s_z|>$). For electrons $s=1/2$ while $s_z=1/2, -1/2$.

Your third sentence is correct. The number quoted in most references is the magnitude of the spin, whereas the "direction" can be positive or negative.

The reason I put direction in quotes above is because it only beomes an actual spatial direction relative to an external field. Spin is an internal magnetic moment that many (but not all) particles possess, and the sign refers to its reaction in an external field: positive spin means it is pushed in the direction of an external field, whereas negative spin means it is pushed in the opposite direction.

In addition, spin need not necessarily be $\frac{1}{2}$; in fact, any integer or half-integer can work. For example, mesons have spin 0 or 1, so they either don't react to a magnetic field or react twice as strongly. Delta particles have spin $\frac{3}{2}$, so they sometimes experience three times the force as an electron.