Can a transformer have multiple, isolated, secondary coils from a single primary coil? Can each coil have a different voltage from the other? If so, is there any large negative effect, and what is the behavior as the load increases?
Yes - transformers can have multiple secondary coils of various voltages. In ancient times, for vacuum tube equipment, power transformers often had three secondaries: 5 volts for rectifier filaments, 6.3 or 12.6 volts for other tube filaments, and a high voltage coil (perhaps ~250 V) for plate supply. The high voltage winding would usually have a center tap (connection to the center of the winding) to allow the use of a full-wave rectifier circuit.
Many transformers have dual primary coils that can be connected in series for 240 volt operation, or in parallel for 120 volt operation.
It is also possible to use transformers "backwards" - using the nominal secondary winding as a primary, and the nominal primary as a secondary.
- Yes, within reason (voltage will be determined by turn count ratio, and the primary will have the same turn count for all of the secondaries, which limits what you can do if the turn counts are small)
- I am not 100% sure about this, but I think it will act like placing multiple single-secondary transformers in parallel.
I have seen setups like this in devices before, namely good quality (HP/Agilent/Keysight) isolated bench top power supplies, that have a single transformer and separate secondary coils to power the GPIB interface board (ground-referenced) and each channel of the power supply (isolated from ground and from each other). Many of these units will also have more complex primaries that can be reconfigured with a switch for efficient operation across a specific set of input voltages (100, 110, 120, 220, etc.)
Multiple secondary windings are very common in isolated DC/DC supplies for IGBT gate drivers. look up pictures of multiple output flyback and IGBT gate drive supply.
As from ur question "a transformer have multiple, isolated, secondary coils from a single primary coil" -Yes Isolation in transformer means simply separation of different windings, From primary to Secondary, even multiple secondary's. Secondary can have multiple voltages 3v,5v,9v,12v,15v,18v etc without any problem. All coils (Secondary) can be used at once or one or as per requirements without any Negative Effect.