Does a headwind/tailwind affect your rate of climb?
A steady (continuous) head wind (or tail wind) will not affect your climb rate, only your climb angle. It means you will reach a specific altitude in the same time interval, but your ground distance will be affected.

A head wind increase (as in a gust) will momentary increase your indicated airspeed, which you can trade for a (momentary) increase of climb rate. Similarly, a tail wind gust will temporarily decrease your indicated airspeed and you might have to pitch the nose down a little, to maintain airspeed and therefore reducing climb rate. These are only transitory effects, until the plane settles back to its original trim speed.