My first guess would be a CPS, crank position sensor. At 10yo or 200k, (which ever comes first) they usually start doing that wacky stuff. It starts sending the computer the wrong information if anything at all. After it cools down, its fine. Most of them get worse with the cool down taking longer and longer.

Usually fuel will sputter but not if the fuel pump gets shut off. There wont be enough pressure to work the system. You need to have someone turn on the key while you are near the fuel pump to see if it spins. They all spin as soon as the key gets turned on to pressurize the system, regardless if it needs it or not. The fun part is that you need to do it when the car is dead..

Do a search and type 'shuts down' or 'won't start when hot'. There are many posts here on that topic.