What would cause the sink to 'cough' when the washer is running?

IF your toilet sink and washer are on the same waste line, which they probably are, the problem is't a short trap. Actually since your washer, toilet and sink have different traps it shoudn't effect anything. The problem probably comes from the washer being what is called a "sudds producing fixture". The sudds produced by your washer can actually go up either your toilet or sink waste line. By code the waste for your washer should be tied in to the sewer main 6' from where your waste line goes up for your bathroom. The only solution is to redo the piping from the sewer main and move your waste line for the washer.

it won't harm anything. The worst thing that could happen is you'll get suds out of the drain in the sink