You could do sonic testing as long as the transmitter and receiver were attached to the object. The transmitter and receiver do not have to be in air to send sound waves, metal or soil will work as the medium too.

If you had 3 Olympic swimming pools full of gold you would have all the worlds gold. Some of the materials we use in modern electronics are of a similar rarity. Meaning that mining asteroids could be economically viable.

Costs for earth mines are amazing. Each tire on those giant trucks was $40k US a decade ago... So a single set of six tyres is a quarter of a million. Mines have heaps of these trucks to mine a couple of grams of gold per tonne of ore. Mining projects already run into the billions. The trucks are already being remote controlled.

Asteroids have several main types mainly carbon types and iron. Biggest issue would be water.