Bluetooth connections do not require a router
Intel
Intel states that Bluetooth connects devices without supporting infrastructure such as routers or access points.
Based on: Intel

No. Bluetooth can connect compatible devices directly without a Wi-Fi network. A Bluetooth mouse or pair of headphones does not need a Wi-Fi router to connect.
Intel explains that Bluetooth peripherals communicate without network infrastructure such as routers or access points. NIST defines Bluetooth as a wireless protocol that lets equipped devices communicate over short distances.
Bluetooth and Wi-Fi can both use the 2.4 GHz band, but sharing a frequency does not make one depend on the other. Their signals may interfere when they overlap in time and frequency.
Bluetooth creates its own wireless connection. Sharing a frequency band with Wi-Fi does not make Wi-Fi necessary.
Intel
Intel states that Bluetooth connects devices without supporting infrastructure such as routers or access points.
Based on: Intel
National Institute of Standards and Technology
NIST defines Bluetooth as a wireless protocol that allows equipped devices to communicate over short distances.
Based on: National Institute of Standards and Technology
Bluetooth Special Interest Group
Bluetooth and Wi-Fi can operate in the same 2.4 GHz band, so overlapping transmissions can interfere.
Based on: Bluetooth Special Interest Group
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No. Microwaves transfer energy to food as heat. They do not make it radioactive, and the microwaves do not remain in the food after the oven stops.

No. Microwaves generate heat within the regions they penetrate. In thick foods, the centre also depends on heat conducted from warmer outer layers.

Yes. Bluetooth takes its name from King Harald Bluetooth of Denmark. It began as a temporary project name inspired by the idea of bringing different groups together.