Any question answered...

I reckon both can be correct depending on whether the speaker is referring to the households, or the selection.

But I'd say the first.

Ditto.

Better to re-write:

Households selected at random are being asked to participate.

A random selection of households is being asked to participate

This. The subject is the singular selection.

"Selection" is a word like family, team, band etc where it can refer to the thing itself (singular) or the individual parts (plural). In some cases either can be correct, depending on what you want to emphasise.

You're referring to the individual parts so it's plural. That's also a nicer sounding sentence.