I can't imagine a relay for dash gauges/idiot lights---a relay is meant to contain very heavy current and therefore is not needed to light up simple bulbs for alternator, etc.
Perhaps you mean "emergency flasher relay"? That might be what you are seeing in the wiring diagram.
Anyway, THAT relay would not affect your problem IMO.
Alternator output at the battery should be something like 13.5V to 14.2 V, more or less.
That red alternator light means that BATTERY current is lighting that light, and that alternator current cannot exceed the battery's 12.5 volts or so. For the light to go out, the alernator has to put out more current than the battery puts into the bulb.
Possibly a bad ignition switch could explain these malfunctions.