Solar battery charging, solar IV curve
1) Yes.  A pv panel connected directly to a battery will be "clamped" at the battery voltage.

2) Yes.  More current at the same voltage means more "power".

3) No.  At the maximum power point, there may be less current, but there is also more voltage and therefore more power.  The trick is to convert the pv's higher-voltage power to the lower battery voltage using a switching dc-dc converter.  Essentially, the extra volts are converted to extra amps.  In other words, a dc-dc switch converter is an integral piece of an MPPT circuit.  Without an efficient power converter, there is no gain.