No oil pressure at idle, but at 2,000 RPM it shoots up to 40 PSI? I rebuilt the engine, could this be the old push-rods and old rocker arms?

Not likely due to bad/old pushrods or rockers since they are just drip lubricated and carry lesser pressure...most oil pressure builds up in bearings for cam and crank. Any knocking noises at idle from low pressure? I'd verify that you have a good working oil pressure sensor and gauge first....could be defective and giving wrong readings. Try a mechanical pressure gauge or tester...sold at parts stores or other sites... you can tap into the factory oil pressure sensor threaded port and see what actual oil pressure you are getting. Should be around 15-20 psi at idle and rise up 10 psi for every 1000 rpms or so. If still low pressure at idle, then likely the oil pump is bad, oil pump relief valve is bad/stuck open or bad/improperly installed engine bearings.

http://www.harborfreight.com/engine-oil-pressure-t...

could be your oil relief valve is open could be you never needed to rebuild the engine the pushrods and rockers will have zero effect on oil pressure , also check the gauge is working you wouldn't be the first person to do a lot of work and find the oil pressure gauge was out

the spiking oil is normal when you accelerate ( it will jump up but go down after a while)

but no pressure at idle is VERY bad.... sounds like your car is either too hot for the oil you have, so its thinning out too much OR you just used an oil that is too thin....

get a heavier weight oil and see if that fixes the problem