We study enhanced single and double ionizations and enhanced single and double excitations in the nonsymmetric two-electron diatomic molecular ion HeH(+) in an intense ultrashort laser pulse linearly polarized along the internuclear axis (z axis). We solve a three-dimensional time-dependent Schrödinger equation, TDSE, via correlated two-electron ab initio calculations within the fixed-nuclei approximation. A complex scaling method is used for calculation of both single and double ionizations. These nonperturbative processes increase with large internuclear distance R and reach a maximum at some critical distance R(c) and decrease by further increase of R. This enhanced ionization (EI) at R(c) is accompanied by enhanced single and double excitation processes. Furthermore, EI is stronger when the permanent dipole moment of the molecule and the electric field at the peak of the laser pulse are antiparallel than when they are parallel. We predict analytically the R(c) at which the enhancement of all these molecular processes happens in HeH(+) from a simple quasistatic model and investigate the effect of Carrier Envelope Phase on these nonlinear nonperturbative processes.