The behaviour you describe is typical of a fully-charged battery in the 50-200Ah range, or one at the end of its life that has very little capacity left to store charge. If the alternator is holding a steady voltage and the current is low, there is absolutely no evidence of a problem with the alt. If the battery were already under suspicion for low capacity I would start out with a heavy current drop-test and/or a hydrometer test on that.
However, to answer your q's:
1a) Correct, inasmuch as varying the voltage will vary the current but the regulator's job is to hold the voltage constant. If it is doing that, it can't further influence the current, only the load can.
1b) No, they are correct under normal working conditions.
2a) It can't really. The regulator works on the field circuit, not the output, so none of the charging current passes through it (unlike a dynamo regulator). If the alternator is fully loaded to the point where its output voltage sags (because vehicle alternators are magnetically self-limiting) then all the regulator does is leave the field running at full strength. Arguably the regulator has less work to do.
2b) The alt windings and rectifier will run hot at full load and if the rectifier is doubtful or starved of cooling that can fail, completely or partly. The symptom you describe does not sound like a bad rectifier, that usually just reduces the maximum output permanently as one or more phases fail, or the whole thing goes short-circuit.
A proper test of an alternator involves running it at full output current to prove that all phases of windings and rectifier hold up. I'd put your machine on the Octopus and pump 40A out of it for 10 mins while it heats up, then make sure it can keep that up at rated voltage at the minimum speed specced for full output. Any less of a load is not a thorough test.
FWIW my boat has (at the moment) four 120Ah batteries and two 150Ah, all charged via relays from a 65A alternator. If the domestic batts are low when the engine is started the alt will run at a steady 65A for a few hours, get toasty hot and the belt can complain a little. After a long day's run its output will be only a few amps above the system load.