Bit error rate curves, theoretical and simulation
Note: usually, the "theoretical BER" is called $P(e)$, the probability of bit error (sometimes with other symbols, like $P_b$). A rate implies experiment; the probability implies theory.

1) $P(e)$ is a lower bound for the BER: since practice is messier than theory, the BER will be higher than $P(e)$ (assuming the theory is correct; that is, that it models reality properly).

2) I'm not sure I understand the question. We always want $P(e)$ to be as low as possible, but it is ultimately bounded by the channel capacity.

3) If by "below the theoretical" you mean "better than $P(e)$", I'd say something is wrong with your simulation or your assumptions.

4 and 5) You use an equalizer when you have a wideband channel with intersymbol interference. Such a system will always have worse BER than a narrowband channel. The equalizer tries to eliminate the ISI; if the equalizer is perfect, the BER of the wideband system will be the same as that of a narrowband channel, assuming equal $E_b/N_0$. In practice, the equalizer will not be perfect. In this case, theory can help predict the equalizer's performance, and how much BER improvement to expect.