The type of diode is known as an stringlass axial glass diode or more commonly , a 'bead' diode.
Unless the 'glass' has turned dark or black, sometimes with these it is a bit hard to tell whether they are good or not. You really have to measure their resistance, in both directions, and then you also have to know the forward resistance value as well to know if they are OK. The forward resistance can be slightly different for the different voltage ratings. The reverse resistance is usually in the megOhms.
I agree with OT03 on the transformer. It looks OK with the brown spot most likely due to heat emanating from a bad winding. Alternatively someone could have accidentally hit the transformer windings insulation with a soldering iron when they were soldering the winding pigtails to the terminal posts. Interesting transformer. By the terminals it looks like there is a choice of two primary windings and a secondary winding with 6 tap offs. But hey that is only guessing and could be totally wrong.