What causes oil dipstick to pop up when trying to start engine?

The only way to know for sure is to remove the oil pan drain plug and drain the oil. Then refill the engine with the correct amount of oil. Then see what the dipstick reads. You might want to invest in the right dipstick tube and stick as one to short will cause you to overfill the engine and one to long will cause you to short the engine on oil and burn it up.

Is it crankcase pressure that causes it to pop-up? If so, check crankcase ventilation system/pcv. It might be possible that you have so much crankcase pressure, that ventilation system can't handle it, then the pressure gets out anyway it can.

On a warm engine, at idle, if you remove oil fill cap, is there smoke coming out?

If you ran it low on oil, there's a sensor in the oil pan that relies on oil to cool, and have been known to fritz.

I would look for other leaks. If you are sure it's leaking at the dipstick tube, there is normally an o-ring where the tube presses into the engine. In general, this is an odd place for a leak, as ther eshould be no oil pressure in the pan (where the tube terminates) and unless the engine is overfilled with oil, the tube end shouldn't be immersed in oil to leak. IF there is pressure in the oil pan, that generally indicates bad rings. I have seen engines where the oil dipstick actually pops out of the tube under pressure when the engine is running due to pressure in the oil pan from blow-by at the rings.

A missing oil dipstick will not cause any issues with your engine.

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If I understand your situation, you are saying the engine was run with one quart and you added 2 more quarts of oil. Now you have 3 quarts which is less than what it takes to make it full. On some engines it takes 5 quarts of oil. You will have to check the dipstick to see where you are now compared to the full mark.

If it is still less than full, it may cause some noise, but depending on how long you ran it with the oil low, how hot it got and other unknowns, you may have a serious problem. Try adding oil to the fill mark and pray that after you run it awhile the noise will go away. The noise may be the lifters trying to work with little or no pressure. It is possible that after the oil top off and a bit of driving they may pressure up and start working again.