Could router logs help find missing Iowa student Mollie Tibbetts?

I'm sure most of you have seen news relating to the girl who went missing in Iowa recently. It was originally believed that she had gone missing while she went for a jog in the area, but the latest news seems to indicate that she may have returned home after the jog and went missing later.

So here's a question for any network experts out there:

I know that our mobile devices, ex. smartphones, broadcast some kind of wifi network search signal/ping 24/7, along with MAC address, as long as wifi on the device is turned on (the phone does NOT have to be connected to the network, but only be within range). The question is, do routers record this information?

If so, could that information be used to determine:

(a) If mollie's mobile device was present in the home, and at what time? (I'm assuming yes this could be determined, since she was probably actually connected to wifi and this is probably logged).

The more curious situation, which could be really helpful, is if routers logs can reveal:

(b) If any NEW (foreign) devices were present in the area of the home, at any point that night. (identifiable by recorded search probes).

(c) In relation to the above, if the search signal/probe/ping from mobile devices looking for wifi networks are in fact recorded, could the routers from nearby homes be used to determine if:

1. Mollie's device was nearby, and at what time 2. If any other devices were nearby at the SAME time as mollie (imagine two devices traveling together down a street, both sending out probes/pings to wifi networks of nearby homes, in tandem together) 3. If a foreign device WAS present near the home she went missing from, and let's say a possible abductor turned her phone off, but left their own phone on -- then they could also perhaps be tracked this way with nearby routers.

It all seems to depend on whether these wifi search/probes are recorded by routers or not.

Anyone know?