Glad that you enjoyed the lecture. In answer to your questions:

1. Yes, the Earth-Moon system will produce gravitational waves! Because the Earth and Moon aren't that massive (compared to black holes say), they won't be too big; you might think they are still important because we are right there... As it is, the waves are at too low a frequency for detection with LIGO: the Moon orbits about one every 28 days, and the ground based detectors are not sensitive to frequencies low than about 1 Hz (one cycle per second).

   However, more directly, the Moon does affect LIGO though its gravitational pull. This is a direct force that can pull at the mirrors at the end of the arms and move them slightly. Fortunately, since we know where the Moon is, we can take this effect into account. In fact, the detectors are so sensitive, that not only must we take into account the Moon, but also the tides. The tides are the movement of water in response the the gravitational pull of the Moon and the Sun. At different points in the cycle there is a different distribution of water across the Earth, and the pull from having extra water in regions close to detectors can be measured.

   In conclusion, the Moon orbitting will produce gravitational waves, but these are not of the right frequency to be detected. However, the Moon's gravity still needs to be taken into account.
    

2. You are quite right that the Sun is changing mass. This change is quite small, at least over human time-scales (the Sun will live for billions of years, so only changes by a tiny amount over a few years, or even centuries). I don't believe that we are currently able to measure the mass of the Sun precisely enough to observe the change; however, this may be possible in the future. Certainly the technology developed for gravitational wave detectors could be adapted to precisely map the Sun's gravitational field. I don't think anyone has come up for a mission design for this yet, although we have recently had missions that have mapped the Earth's and the Moon's gravitational fields (GRACE, GOCE and GRAIL; a GRACE II mission is planned, and will hopefully include technology originally designed for LISA).

   Regarding gravitational wave measurement, you've hit a very important point. There shouldn't be any (significant) oscillation. To excite gravitational waves you must have something like a binary (technically where the mass quadrupole is changing) and not something spherically symmetric. The Sun's mass loss is very nearly perfectly spherical, and so there shouldn't be any real gravitational wave emission.