Hello again jm129852,
Having two power and ground terminals allows you to run two smaller wires instead of one big one. It's a convenience. No, you do not have to use them both. They are connected together inside. Yes, you should still use a single fused battery lead. Connect the big wire and fuse holder to the battery and to a distribution block and run the smaller wires from the output side of the distribution block.
When you connect the outputs of a two channel amp to two separate speakers, each speaker gets the power produced by it's channel only. If you are using only one speaker, and the amp is rated to operate bridged, then the speaker will receive the power from both channels, typically twice as much. You do need to be careful when using bridged mode because many amps will only operate safely at a higher impedance when bridged. For example, when operated with both channels connected to separate speakers (not bridged), it may be stable to 2 ohms. But when you connect both channels together (bridged), it is only stable to 4 ohms. Usually, the specifications will tell you the lowest impedance at which the amp should be operated. The specifications for the current RF Punch 500.2 are like this: 125 watts X 2 at 4 ohms; 250 watts X 2 at 2 ohms; 500 watts X 1 bridged at 4 ohms (4 ohm stable in bridged mode). You would NOT want to connect two 4 ohm subs in parallel to this amp in bridged mode. It would result in a final 2 ohm load and the amp would overheat and fail. One 4 ohm sub would be OK, and it would receive the full 500 watts as stated in the specifications. You could connect four 4 ohm subs, two each in parallel to each channel, and the 2 ohm loads on each channel would be OK. Or you could series two of them together for an 8 ohm load, do the same with the other two, and then parallel the two 8 ohm loads for a final impedance of 4 ohms and connect them into the bridged terminals. With DVC subs the connection possibilities get more complex. There are many good impedance calculators
online that can help you determine the best wiring solution for various amp/sub combinations. I like the one above at the12v.com web site. Rockford-Fosgate also has one which allow you to select the a number of subs (up to 4), their voice coil configuration (single or dual), and the voice coil impedance. The RF calculator then shows you what configurations are available and what your final load impedance will be.
Hope this helps.