You really need tools like and Oscope with a differential probe, Sencore LC102 is good at checking caps, coils, flybacks, transformers. BK520B for checking Transistors, Diodes. Curve Tracer for checking HV diodes, Zener Diodes, SCR's
Otherwise it's a case of shotgunning parts. Starting with the most common stuff to go wrong, like Caps, Diodes, PWM, SCR's Transistors.
100mH 5A SF Coil
This will check as basically a short. They rarely go bad, so unless it's burned up it should be good.
3D-1001 Hybrid IC
That may be a tough one, are replacements even available?
2SC2535 Silicon NPN Transistor
This explains it better than I could.
http://www.allaboutcircuits.com/vol_3/chpt_4/3.html
3D-0097 EI-35
O scope I assume would be the best way.
Looking at the parts you've listed, I'm assuming this is a Nitendo PS?
On Switch mode supplies, Electrolytic caps are the #1 problems.
They can cause things like, not starting, not holding up under load, to much AC ripple on the DC out.