Determination of +/- reflections in sigma molecular term symbols
The $ \pm $ superscripts refer to symmetry operations defined in the $D_{\infty h}$ point group.  The operation on the molecular orbital is reflection in a plane containing the nuclei. If the orbital in unchanged then the sign is $+$, if changed then $-$. 

The electron configuration for ground state oxygen is 
$(1\sigma _g)^2(1\sigma _u)^2(2\sigma _g)^2(2 \sigma _u)^2(3\sigma _g)^2(1\pi _u)^2(1\pi _g)^2(1\pi _g)^1(1\pi _g)^1$

However, the only electrons we need to consider are the two in the $\pi_g$ orbitals. Adding up the individual orbital and then spin angular momenta separately (equivalent to Russell-Saunders coupling in atoms) then combining both gives the term symbols $^3\Sigma, ^1\Delta ,^1\Sigma $. The calculation is done in a similar way to that for atoms and is shown below in the notes.

The subscript g is found because there is an even number of $\pi_g$ electrons.
The superscript $ \pm $ is because any state ( singlet or triplet) must be asymmetrical overall to exchange and as the triplet has a symmetrical spin wavefunction the spatial part is asymmetrical, and vice-versa for the singlet, the superscript for the triplet is $-$. Thus the superscript gives the term symbol $^3\Sigma_g^-$. Hund’s rule dictate that this is the lowest energy level, followed by $^1\Delta$ at $7918 \pu{cm^{-1}}$ and $^1\Sigma_g^+$ at $13195.22 \pu{cm^{-1}}$.

The $\ce{O_2^-}$ has one unpaired electron in a $\pi$ orbital (as does $\ce{O_2^+}$) so the term symbol is $^2\Pi_g$.

As to your third question the overall symmetry should not change when you go from atomic to MO's.

Notes 

The individual $m_l$ and $m_s$ are listed below. The values restricted by the Pauli principle are removed.
The $m_l$ values are $\pm 1$ and the $m_s \pm 1/2$. The the table the spins are labelled as $\pm$ only. The total angular momentum is given as $\Omega = \Lambda + \Sigma$ by analogy with $J=L+S$ in atoms.

$$\begin{matrix}
m_{l1} &amp; m_{l2} &amp; m_{s1} &amp; m_{s2} &amp; \Lambda &amp; \Sigma \\
1 &amp; 1 &amp;  + &amp; + &amp;\text{etc.  Pauli forbid'n}\\
\\
1 &amp; 1 &amp;  + &amp; - &amp; 2 &amp; 0 &amp; ^1\Delta\\
1 &amp; 1 &amp;  - &amp; + &amp; 2 &amp; 0 &amp;  .. \\
-1 &amp; -1 &amp;  + &amp; - &amp; -2 &amp; 0 &amp; ..\\
-1 &amp; -1 &amp;  - &amp; + &amp; -2 &amp; 0 &amp; ..\\
\\
1 &amp; -1 &amp;  + &amp; +  &amp; 0 &amp; 1 &amp; ^3\Sigma\\
-1 &amp; 1 &amp;  + &amp; +  &amp; 0 &amp; 1 &amp; ..\\
1 &amp; -1 &amp;  - &amp; -  &amp; 0 &amp; -1 &amp; ..\\
-1 &amp; 1 &amp;  - &amp; -  &amp; 0 &amp; -1 &amp; ..\\
\\
1 &amp; -1 &amp;  + &amp; - &amp; 0 &amp; 0 &amp; ^1\Sigma\\
1 &amp; -1 &amp;  - &amp; + &amp; 0 &amp; 0 &amp; ..\\
\end{matrix}$$