Exact parameters are based on specific requirements of your application. If you don't have anything specific, "reasonably good results" is fine. Remember that you make trade-offs between noise suppression and sensitivity to DC component variations. When both are acceptable, you're done.

Yes, indeed. Corrected.

I thought this was about making a guitar synthesizer, this topic is interesting to me and I thought maybe you're familiar with it. "Plucked string" is also known as Karplus–Strong string synthesis algorithm and is relatively easy way to synthesize a string-like sound. Buffer is filled with random samples and then repeated while continuously applying low-pass filter to it. Length of the buffer determines base pitch. Sometimes it's referred as "physical modelling synthesis". I thought maybe it's possible to achieve more natural results by combining it with physical model of a guitar pick-up.

Is "plucked string" algorithm a reasonable choice for the original waveform generator?

You can assume $x_n = 0, 1,\ldots N-1$. After you find the approximation $y_n = t_d \cdot x_n^d$, you may find $x_n' = \frac{x_n}{t_d}$. This will give you $\theta = {x'}^d$.