Start over.
Check the amperage output of the battery. I don't know how you've been checking it but a battery can show 12 volts and still not provide enough amperage. Also check available amperage through the engine block and intake manifold. This will tell you if you have a bad or weak ground due to loose anchor, corrosion on an anchor or corroded cable within the insulation that is hampering ground conductivity.
Ground points are: on a bracket just above the upper starter bolt, left fender by the battery, on the intake manifold passenger side, straps that run from the engine to body/firewall, strap to the ABS assembly.
If the battery checks good, amperage output above 400, the higher the better, then check the alternator with a load on it.
You can check the fuel pump flow with a hose and a gallon bottle/bucket and a stop watch. The pump should flow upto 55 gph with no restriction, low end about 35. It can be toggled on via the fuel pump relay by jumping pins 30 and 87.
Fuel pressure can only be checked with a fuel pressure gauge, which is available at most parts stores in simple fashion that will work on your car. It should be approximately 43 psi at idle and increase with the FPR vacuum hose pulled. But you've already changed the FPR and hopefully from Subaru, so you are just checking the output of the pump.
Vacuum leaks will be tricky without a "smoke machine". You'll have to listen closely or use an accelerant to spray at vacuum connections, hoses and along the intake seals and gasket areas.
I've reread through the chain of events in this thread and a few things stick out. The lack of power then kicks in is an indication of one of the above listed causing the issue. The misfires can be caused by one of the above. A vacuum leak especially will cause what you are experiencing as a sudden change in pressure by opening the throttle, the ECM increasing the injector pulse to comply with the request by your right foot, the altercation of the timing to fire the cylinders properly to increase engine speed will all be thrown off by a vacuum leak. When there is a leak, it throws all the math off. As the engine rpm finally rises, the ECM has recalculated based off the input it gets from the AF sensor, and is trying to keep the engine running.
A bad battery will cause low amperage output and also effect the alternator output. Low amperage = Stupid ECM. It won't be able to function and control the actuators on the car with low amperage. It also follows that as the engine rpm gradually increases, the alternator output will bump up and provide more amperage for the ECM and actuators to function.
Low fuel volume to the cylinders during compression will cause detonation, pinging. If you are getting detonation, the air/fuel ratio is off either due to a bad pump, clogged filter, vacuum leak or poor timing. As the engine rpm increases, the ECM will increase the fuel injector pulse and alter the timing based on AF sensor and knock sensor feedback while calculating the TPS, ECT and MAF.
And again, this is based on what you have posted, but it sounds like its either a bad battery or a vacuum leak. I doubt its the fuel pump. Its possible, but on a low scale based on what you have posted.
If after checking the battery/alt, vacuum, cabling, fuel flow and pressure and you are sure there are zero issues with any of these, then move to the sensors starting with the MAF.
Good Luck.
