According to Molecular Orbital Theory (MOT):

1. Magnetism of $O_2^+$: The oxygen molecule ($O_2$) has 16 electrons and is paramagnetic because it contains two unpaired electrons in its $\pi^* 2p_x$ and $\pi^* 2p_y$ antibonding orbitals . When $O_2$ loses an electron to form the $O_2^+$ ion, the electron is removed from one of these antibonding orbitals, leaving one unpaired electron behind. Since species with unpaired electrons are paramagnetic, $O_2^+$ is paramagnetic, not diamagnetic . This makes Option 1 incorrect.
2. Bond Orders: Bond order is calculated as $\frac{1}{2}(N_b - N_a)$ . For $O_2$ (16e), $BO = 2$; for $O_2^+$ (15e), $BO = 2.5$; for $O_2^-$ (17e), $BO = 1.5$; and for $O_2^{2-}$ (18e), $BO = 1.0$ . This statement is correct.
3. $C_2$ Molecule: $C_2$ has 12 electrons with the configuration $(\sigma 1s)^2 (\sigma^* 1s)^2 (\sigma 2s)^2 (\sigma^* 2s)^2 (\pi 2p_x^2 = \pi 2p_y^2)$ . It indeed has four electrons in its two degenerate $\pi$ molecular orbitals. This statement is correct.
4. $H_2^+$ Ion: A neutral hydrogen atom has one electron; a $H_2$ molecule has two . The $H_2^+$ ion is formed by removing one electron from the $H_2$ molecule, leaving it with exactly one electron . This statement is correct.