Daisy Chain 120V Disconnect Box from existing 240V box

It looks like the cable was indeed originally run for a 220V stove.

When the stove was replaced with a gas stove, rather than run a new cable they just used the existing cable. This is fine as long as both ends are properly terminated.

I don't like the way they terminated the outlet though. I think they should have used a 12 AWG pigtail. ( assuming it was a NEMA 5-15 or NEMA 5-20 outlet ) Other than that I might have done the same thing depending on the difficulty of running a new cable.

( NEMA 5-15 is the designation of a US 110V 15A outlet )

If it is easy enough, I would run a 12 AWG cable from the breaker box to another box near the first with a NEMA 5-15 or 5-20 outlet.
Then either put an appropriate outlet on the existing cable with a 40A double pole breaker (assuming it is 8 AWG wire); or just put a blank cover on it.
If you do just use a blank cover, you should either ground out all of the conductors or remove the cable from the breaker box.
This way it can be left wired for both gas and electric stoves.

If it is difficult to run a new wire, I would just do what they did; except use a pigtail to connect the outlet. Don't cut back part of the wire, use an appropriately sized wire nut.

Both of the things you report - the smell and the hum - are very likely indications of a bad connection or bad contact in the disconnect.

Bad connections and bad contacts generate heat, the heat can damage the insulation of the wires, and even cause a fire.

I'd leave that disconnect off and if possible turn of the breaker for the circuit until it can be checked. You'll probably wind up having the disconnect replaced, hopefully the wiring can be re-terminated without replacing it.