Question 1 (biasing): It is no surprise that the output signal contains a dc part because the input signal is centered upon a dc voltage. However, the dc ouput level cannot be verified simply by amplifying the input dc voltage because the form of the output signal has changed.
Question 2 (form): There are two different effects which influence the output form: (a) Limited and frequency-dependent small signal gain of the opamp and (b) limited slew rate of the opamp. The slew rate may be the reason for the observed behaviour in case of a wideband opamp model.
EDIT 1: Simulations based on the LF411 model show pretty good results (as expected): Output 2.3Vpp (between 0.7 and 2v) with slight distortions and a very small noise component (5 Mhz) riding on the top of the signal.
EDIT 2: I think, both questions are answered: DC portion and signal form distortion.