The rectifier and regulator on this type of alternator are separate devices, I don't know what this white wire is for. There is no direct connection required between the rectifier and regulator.
A 3 phase rectifier has 5 terminals: 3 AC ones, usually marked with a "~" symbol, and the DC output marked "+" and "-". The three yellow wires from the alternator stator coil connect to the AC terminals, "-" goes to a solid ground on the frame or an engine mounting bolt, "+" goes direct to the battery "+".
The alternator is a controlled field type. The field coil is an electromagnet; the regulator varies the field coil current to control the alternator output. The regulator usually has 3 terminals - one ground, one for the field coil, and one connected to switched ignition power. The field coil obviously needs two wires, the one not connected to the regulator is either grounded or connected to +12V depending on its type. I believe all OEM SOHC4 alternators (excepting GLs, a different story entirely) connect the other field wire to ground.
The rectifier itself creates heat when operating. It needs some sort of heat sink, the integrated module type I think you're buying will have a mounting hole to bolt it to a heat sink. You don't need a huge heat sink but you need something, many owners use a surplus store computer chip heatsink from an old computer like a 486 or early pentium.
The original regulator should work fine, but there are many options for replacements. Newer Honda electronic regulators from a DOHC4 work if you connect the other field coil wire to +12V.
The 550 alternator uses a powered field coil in the rotor, with slip rings and brushes to get power to the spinning rotor. The brushes need to be serviced occasionally and the rotor itself is a bit problematic - they are subject to a lot of vibration and acceleration force and they fail on occasion. I can't recall the exact resistance you should measure between the field coil wires but it should be checked if you're having a charging problem.
