Will this circuit work for smoothing out DC adapter output?
1)to be honest, I don't think the fluctuations would be causing that. Batteries themselves cause bigger issues when their own voltage drops with normal usage. I do not know the levels of your equipment though. It would be helpful if you mentioned what kind of batteries it needs.

2)the circuit you write will work, but the value for the capacitance is rather huge, as someone commented on your post for that voltage value it is a problem, and having that much of a capacitance might cause other issues(voltage might go above what you specified depending on the situation). I think a not so huge capacitor still can get the job done.

3) honestly 1N4148 should work, I don't know the voltage and current specifications exactly, but just look at the datasheet online and compare if your reverse voltage, current, etc do not go over the parameters, also consider that when current is going through the diode there will be about  0.7V  between its terminals(ideal value, not exact)

4) I don't think it is related. 

I have one thing to propose, I don't know what is the actual input voltage of the faucet with batteries. For example, if it uses 4 AAA batteries it could be 6V if they are in series, try to match this voltage the best you can. I guess since your circuit says 10V it could be close to that? If your adapter does not match you can use DC/DC converters or buy an adapter that is closer to the value you need. I  believe this is most critical in general.