If these components are on the secondary side of the power supply, then they have to be Schottky diodes because they need to be fast. This power supply is running at tens of kilohertz, and a 1N4007 just won't do the job: it recovers much too slowly. You need to get a Schottky diode for this position. It doesn't have to be exactly the right one. Anything of adequate current and voltage rating will do the job. A scrap PC power supply is a good source of such components. The 1N4007 will not work, I guarantee you.
The rectifier circuit isn't just a half-wave rectifier. If it's the standby supply, it's probably a flyback converter. There's a useful explanation on Wikipedia here:
https://en.wikipedia.org/wiki/Flyback_converter
The magnetic field in the transformer is not a sine wave, and each cycle is divided into distinct parts during which the diodes, transistors and inductors play particular roles.
A transformer that size is perfectly capable of delivering 10-20W, so a couple of amps at 7V is no trouble. Because it's running at a high frequency, it's much smaller than a 50Hz mains transformer.
Chris