First off, nice job with the probe, the signal looks OK for this kind of set-up. Really Impressed. Looking forward to seeing this thing in action

1.Maybe play with the op-amp gain and/or with the input attenuation resistor network.
2.Well, basically, yes, it's a must. Your signal on the output will depend , among other things, on the slew rate of the op-amp. No problems with the one you chose, but once the batteries will get drained, it will decrease in performance. Also, it will have an effect on the amplitude of the output signal. So, maybe stick an LDO on each rail and get things as stable as possible. Another way would be to get a buck-boost power supply, powered from the batteries, to get things stable even when the batteries are really low. but that requires some shielding of the PSU portion from the rest of the board.
Also, have a look at the graph where it says "power supply rejection versus frequency", that will give you some idea of how the well the op-amp rejects noise picked up from the power supply, because that will also influence you final waveform output.
3.Yes and no. Yes, you always have to connect the "+" and "-" of the probe to the DUT. No, the GND of the probe is never connected to the DUT. That goes to the oscilloscope GND. Because it's a differential probe, you measure a signal differentially i.e. wherever you connect the "-", that's going to be you reference (you can hook up the "-" to the GND of the DUT or to any other part of the circuit, given the probe can withstand that much voltage difference). If you connect the GND of the Diff Probe to the GND of the DUT, you defeat the whole purpose of a diff. probe.
4.It depends. If you'll be measuring DC level voltages (0Hz to about 50KHz), then the wires are perfectly OK. But if it's AC measurements you;re interested in, asyou have showm in your photo i.e. the 1MHz signal, then yes, you need some proper screened cable, in order to limit the amount of noise pick-up and also get a controlled input impedance on the input of your probe.
Don't worry, the questions are not stupid, it's how people learn stuff

As a side note, the op-amp you've chosen can handle really high bandwidth, but you might want to look for other methods of building you Diff Probe, because an op-amp in differential amplifier mode has severe limitations as far as CMRR is concerned. Also the bandwidth will be much less then what those fancy charts show in the data sheet.
Also, it's always better to buffer you input signal (if you want to have a bandwidth above 1MHz) with some really high impedance, something like a JFET differential stage. The input impedance of resistor network and the op-amp alone will not cut it.