Secondly, the internal HD seems to behave oddly: sometimes it doesn't spin up at all, some other times it spins and shows an happy Mac - only to return to the blinking floppy a few seconds later. Is it dying?

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The Miniscribe drives used in the SE are prone to stiction which causes the drive to not spin up. A brief happy mac followed by the blinking question mark again is normally an indication that the system software is damaged.