Integrator with DC component
You have a DC source trying to drive it positive relentlessly and quickly towards the positive rail, so it spends some time saturated there every cycle. When the 400Hz AC input overcomes the -400mV source it struggles off the positive rail and integrates resulting partial half cycle.
Integration, if you recall your basic maths, always has a constant. You've also got a constant output slope of +4kV/s from the -400mV source. The op-amp also has limited range of output. If the op-amp was ideal, your output would shoot up by +80V in the shown 20ms from the -400mV constant alone, and you'd get the cos wave on top of that, plus some constant depending on initial conditions. It's not ideal, and the op-amp railing essentially resets the integrator every cycle, but at the positive rail, not at 0V.

3./4./5. It's nonlinear because you're hitting the rails.