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As for your question about why the charge voltage was higher with load, it was probably a function of time, rather than load. If the batteries were not fully charged initially, they would have done a pretty good job of regulating the charge voltage, since the alternator puts out only a few...
Any of the 1N400X series (1N4001, 1N4002, etc.) will work fine. The 4001 is rated at 50V, the 4002 100V, on up to the 4007 at 1000V. All are 1 amp diodes. They are all inexpensive and available from the usual mail order suppliers. Radio Shack used to stock them, but I haven't checked them in...
Very true, but many meters have difficulty measuring the resistance (ohms) of a diode. Some have a setting specifically to test diodes. A simple battery and test lamp can test them just as well, once one end is disconnected from the circuit.
I know of no reason running this generator without load would hurt anything. Larger water cooled diesels can develop excessive deposits in their exhaust systems when run lightly loaded for long periods.