Is this a plausible method of extracting useful amounts of energy, or are we talking about yoctowatts per cubic yottametre?

I guarantee you that these rectennas will not work if their temperature is equal to that of the source radiation, and that they will generate waste heat which has to be removed. As for slaves to the Carnot cycle, no one has ever built a Carnot cycle; it is merely an ideal which these rectennas might help us approach more closely.

Most likely before "free energy for all" you'll have strange IR oriented sensor apps that can analyze the spectrum not just measure total power.

Personally I think it would be fun to see a revival of IR spectroscopy outside of o-chem labs. Its been mostly superseded by NMR in the labs. But imagine an IR spectroscope on a chip. Perhaps for med purposes (what?) or maybe on oil well drilling down hole analysis or something like that.

Or just day to day appliances, like a "smart" household smoke alarm that analyzes the spectrum to identify and ignore tobacco and broiled/grilled meat combustion products but gets really excited about any measurable combustion products of burning furniture or burning paints or burning fabrics. I bet you could cut minutes off fire detection time with one of those, which doesn't sound like much but I bet it would save a lot of lives.

This is bullshit (specifically, a perpetual motion machine of the 2nd kind).

Suppose you have a heat source T_H in an environment T_C. If you could extract more work from the thermal radiation of T_H than the Carnot limit for this system, you have enough work to run a Carnot heat pump from T_C to T_H that increases the temperature of T_H, and decreases the temperature of T_C -- an violation of the 2nd law. You could use work to increase the heat source's temperature, leading to more work, leading to higher temperatures, in perpetuum. No can do!

It's far from just colloquial. Infrared and thermal motion at the same temperature contain the same amount of exergy, meaning that the thermodynamic limits on how much work they can do are the same.