How to send / receive a short impulse in an extremely noisy signal with high timing accuracy?
Using a sinusoid is a bad idea. In order to precisely detect the time of arrival, you want a signal with wide bandwidth. Here's a brief description of a good approach:


Transmit a known direct-sequence spread spectrum (DSSS) waveform (typically a phase-shift-keyed digital signal, either BPSK or QPSK). You should pick a spreading code that has good autocorrelation properties: its autocorrelation should look close to an impulse. A sequence of random symbols will often work, or you can use something more well-defined like a maximal-length sequence.
At the receiver, apply a matched filter to the known transmitted pattern. 
You will see a spike in magnitude at the output of the filter that corresponds to the time of arrival of the signal of interest. Apply a threshold to the filter output to identify the locations of these spikes.
The matched filter will have a delay of $N-1$ samples, where $N$ is the length of the sequence. Subtract this duration from the detected pulse locations to yield an estimate of the pulse arrival time.


Optionally, take the samples surrounding the peak and interpolate to get a finer (sub-sample) estimate of the actual peak location. Polynomial, sinc-based, or other methods may be used here.



This is a very simplified description of what you might want to do, but it should get you started. One consideration to keep in mind when designing the length of the sequence: there is a tradeoff between the total integration period (i.e. the sequence length) and the frequency selectivity of the matched filter. If you have appreciable frequency error between your transmitter and receiver, the peaks at the matched filter output will become attenuated. The amount of attenuation depends on the frequency offset's relationship to the matched filter length. This is a common attribute of DSSS systems that must be handled appropriately.