That is your EGR Vacuum Solenoid/EVR(egr vacuum regulator)Solenoid making the sound after you cut the engine off. I have heard that "tooting"
sound on my ST as well. I don't believe it to be any kind of issue. I have tested my entire EGR system. All good.
It's the item with the round cap (tiny hole in center, 6 "teeth" and an electrical connector) on it in between the EGR Valve(thing with the vacuum nipple on top) and Coil Pack.
Try giving it a spin. It spins freely. It may quiet it down or change in tone. That worked for mine IIRC or it just stopped tooting by itself. I cannot recall, sorry. I just ran outside and checked. It does not toot any longer but it definitely makes a slight sound/vibration 2 seconds after shutting the engine down. Barely.
That being said...my ST starts and runs perfectly and I don't believe this is your issue. BUT, a faulty item in the egr system can cause stalling and rough idling. (not so sure about a cranking no start though) If you want to test the entire EGR system(egr passages, dpfe sensor, evr, wiring, egr vacuum hoses, and pcm driver in one easy test, watch this video.
Video: Ford EGR Flow Testing by ScannerDanner (if you don't have a scan tool...t-pin top wire(signal wire) on dpfe sensor to read dpfe voltage on a volt meter while doing the test)
(actually watch ALL his EGR videos. Search "ScannerDanner EGR" They are great! IF you watch all of them you will be a EGR master by the end)