Do I understand correctly that the only problem you see is that the unit displays random readings with no input? If so, this is perfectly normal. The 300mV and 3V ranges on this meter are high-impedance, much higher than the usual 10Meg. The input will just accumulate charge from the environment and show readings until it reaches 3V, at which point the unit will auto-range up, which will dissipate the charge, and the process will repeat.
If you short the inputs and it reads zero plus/minus a few counts, then it is working normally. If the random readings bother you, obtain or make a shorting plug and leave it in the jacks until you use the meter.
As for your scope waveforms, no they don't exactly look normal. Using a X1 probe maybe? Hanging loads on crystals can kill or alter their oscillation. At least the ALE signal is about the right amplitude and frequency. You're probably seeing the limitations of the pocket scope.