Question

1. What can you say in general about the effect of wire/string thickness in some instruments on the produced frequencies?

Answer 1

Gnerally the strings which can produce the frequencies on any instrument are fixed at both ends.
the standing waves will form on them due to the incident and reflected waves supeposition.

the relation between the frequency wavelength and the speed of the wave is

   v = Lambda*f

the speed of the waves on strings depends on the nature of the string, is given by

   v = sqrt(Y/rho)

where Y is young's modulus of the wire and 'rho' is linear mass density.
here, the string or rod expands sideways slightly when it is compressed longitudinally, it is possible,only the string or rod whose sides are free to bulge and shrink a little as the wave travels, it may not be possible for the rods and strings which are like bulk solid, since in these materials sideways motion is prevented by surrounding material.

   so frequency is  

and again the wavelength of the wave on the string depends on the length of the string (depends on the mode of vibration producing corresponding frequency)