Hi,
Running the diode at full rating probably isnt a good idea.
And yes, if the diode is rated less then it's voltage drop at a given current would normally be higher because the lower rated diode has less cross sectional area, just like what would happen with a resistor of less cross sectional area but made with the same material. a resistor with 1/2 the area of another resistor made from the same material and same length would have 2 times the resistance and thus 2 times the voltage drop for a given current.
The 1N5822 isnt the best choice either though, even though i have used it in this very same application. That's a 3 amp diode as you know, and with a 3 amp output the diode rating should be more like 4 amps. That's if you use the full 3 amp output rating though. For my app, the current was LIMITED via a secondary transistor circuit that limits the current to 1 amp so the output can never go above 1 amp. I dont know if you have that kind of circuit but i suspect that you dont. Thus your circuit depends entirely on the behavior of the chip alone.
I cant remember if this chip has slow start built in or not. If it doesnt, then the initial current could be high because the cap is discharged when the circuit is first turned on.
Interesting that you should bring this 1 amp diode question up here because i've often thought about what would happen with just a 1 amp diode in a 1 amp application. To be more accurate though, i would either find a diode that is rated for 4/3 amps or limit the current to 3/4 amps if i used the 1 amp diode.
The diode normally has to deal with the max current in the inductor, and we think that is 3 amps, but it can actually be higher and that pushes the RMS current higher and thus the power dissipation. I think that is the reason for the 4/3 rating factor requirement although it's been a long while since i looked at this circuit now.
If you throw a 5 amp diode in there it will probably run forever.