Why does the toilet have two overflow tubes and a refill line that tees

The most common problem that causes the symptoms you describe is a slow leak around the flapper. A very small amount of water drains into the bowl over time, lowers the tank level a slight amount and activates the float valve of a couple of seconds to refill the tank. What you should do before getting carried away dissecting the fill valve, is to check the flapper, clean it and it's seat well and be sure it is centered on the drain. If there are any signs of wear or cracks around the edges, replace it. Monitor the tank level between phantom fills, does it change at all? The second thing is to slightly lower the float so the water fills to apx 1/2 inch below the fill mark. This will eliminate the possibility that you may be losing water over the overflow, and tripping the fill valve.

These checks are simple, cheap and often solve a very common problem. Good Luck.

That's a fairly popular fill valve. To open it:

1. Shutoff the water, you don't want a geyser.
2. Lift the float to the top position.
3. While holding up the float, push down on the top cap and twist the cap a quarter turn (I believe counter clockwise).

The cap will then lift off and you can inspect and clean any parts. Before reinstalling the cap, move any small parts to a safe location, get a small cup to place over the top of the valve, and turn on the water supply. This will jet water straight up from the top of the valve and clear out any debris.

Reinstalling the cap is the reverse of the above procedure. Put the parts back, position the cap back on top, lift the float, press down on the cap, and twist.

Note that if cleaning it doesn't work, it's fairly simple to replace this fill valve.