Linear power supply loading question
With no manufacturer, model number, photos, or schematic, all answers are guesswork.  The true answers to the first three questions are maybe, maybe, and maybe.

4 - Yes.  In your case, it sounds like a foldback current limiting circuit hysteresis.

2 - Most switching power supplies will power up into a 100% load.  Depending on the size and complexity of the supply, it might have a "soft-start" design that ramps up the output voltage to reduce the startup input current spike.

1 - Maybe, but it is not an common industry practice.  Not that if the downstream load has a significant input capacitance, this appears to the supply's output stage as a brief dead short.  This probably is going on in your situation, but without any information about the load ...