A Dyson's sphere is a device to convert high frequency photons (visible light and uv) to low frequency photons (radiated “heat”). A sufficiently deep stack of shells can bring the temperature of the radiated light closer to the temperature of the cosmic background radiation, but it absolutely will radiate.
The question isn't whether Dyson speheres radiate, the question is can we detect an artificial megastructure and my answer is no, based on the hypothetical Dyson design.
That is physically impossible unless there is new physics in that hypothetical design. All physical objects radiate heat and a Dyson sphere in particular would be trivial to detect. You look take a picture of the sky in infrared and in the visible spectrum. If you find an infrared source but no associated visible star you’ve got a strong candidate for being a Dyson sphere. Such searches have actually been conducted.
Other megastructures might be discovered through the same methods as exoplanets.
Yes, but a Dyson sphere, even around a red dwarf, would be far more luminous in the infrared than a brown dwarf. It would also have a different spectrographic signature, and importantly its heat distribution would appear artificial.
