Hulse-Taylor binary pulsar - what is the rate of mass/energy loss from the source?
For reference, the relevant bit from the paper is:


  The observable pulsar is a weak radio source with a flux density of about $1\,\mathrm{mJy}$ at $1400\,\mathrm{MHz}$. ... Our most recent data have been gathered with the Wideband Arecibo Pulsar Processors (“WAPPs”), which for
  PSR B1913+16 achieve $13\,\mathrm{\mu s}$ time-of-arrival measurements in each of four $100\,\mathrm{MHz}$ bands, using $5$-minute integrations.


Those are millijanskys, aka milli flux units, so that the flux density is about
$$1\,\mathrm{mJy} = 10^{-29}\,\frac{\mathrm{W}}{\mathrm{m}^2\cdot\mathrm{Hz}}\text{,}$$
and hence the detected irradiance is on the order of $10^{-27}\,\mathrm{W}/\mathrm{m}^2$. Since we're about to make rather uncertain assumptions anyway, I won't bother worrying about doing more than an order-of-magnitude calculation.


  Assuming that this flux is uniform across a conical beam with cross-sectional radius 5 arc degrees and assuming that the source is 21,000 light years from Earth ... - How much energy is emitted (per second) from the source in the beam?


A spherical cap has surface area of $A = 2\pi Rh$, and here $R = 21\,\mathrm{kly}$. Now, I'm unclear what cross-sectional radius means if measured as an angle, but I take it to mean that the opening half-angle of the cone is $\frac{\vartheta}{2} = 5^\circ$, in which case
$$A = 2\pi R^2\left(1-\cos\frac{\vartheta}{2}\right) \sim 10^{39}\,\mathrm{m}^2\text{.}$$
Thus, the power would be $P\sim 10^{12}\,\mathrm{W}$, but note that in addition to the assumptions you've just listed, we're only talking about a particular radio band.