1. The reasons for using filament transformers with 12 AC output is as follows. The filament transformers' power (in this case 30W which is calculated as W = V x I) has to be such so it could supply a clean AC voltage to Trident and subsequently supply clean DC voltage to Frigate. The current that flows through each filament bus of Trident and Frigate is 1.5 A. This is a substantial current. Selecting transformers' AC voltage lower than 12 AC would result in insufficient current produced by the transformers and insufficient DC voltage produced by Trident. Since power is calculated as W = V x I, to sustain the current required by Frigate, the voltage would drop to the lever below 6.3 VDC and you would hear ripple in your headphones. Also, for LD1084 to give out at least 6.3VDC, they would need to receive at least 9.0VDC in. As you can see, taking in consideration all the above, the minimum output AC voltage for the filament transformer should be 12 VAC.
You measured filament DC voltage without any load connected to the Trident. Once you connect Frigate to the Trident, the voltage will drop to 6.3 - 6.6VDC.
2. The heat sinks of all three voltage regulators have to be hot. That is the reason they are placed on heat sinks. The current that floes through filament voltage regulators is 1.5A and the current that flows though the B+ voltage regulator is 0.1A. These are substantial currents and therefore the regulators become hot. Both transformers, filament and B+, should be warm with temperature staying steady at about +35 C.
3. Some customers buy Frigate without power supply Trident and they use their own power supplies. This is not recommended but this is their choice. Since we do not know what voltage their power supplies would produce, we have shown on the Frigate's schematic that the B+ voltage has to be +250 VDC. The Trident power supply provides about +280 VDC without a load and +250 VDC with a load.
Please post photos of your Frigate and Trident assembly.