If BEagle's clock-code system isn't scientific enough for you, what you're doing to determine the crosswind component is multiplying the sine of the angle of the wind by its speed. To do this, you need the sine of that angle. The clock-code is plenty accurate enough for determining the sine of that angle and better still errs on the side of caution at the greater angles.
For the first question - the calculation at takeoff is pretty easy using that clock code. 15deg wind - a quarter of the wind speed is crosswind. 30deg wind - half the wind speed is crosswind. 45 its three quarters. Just these three alone are accurate enough for most purposes I would have thought.
For the second question, use BEagle's method again. In a training aircraft doing around 110 knots I'd call it 2 miles a minute, so halve the windspeed and apply the clock code.
As for how you get the windspeed (before you're even tracking a beacon) if you're at 2000' you can take the surface wind from ATC, double it and add 30deg (in the UK) for a rough estimate.