What is the (approximate) function for amplitude of a plucked string over time? Does it differ between string types?
An exponentially decaying envelope $a\exp(-b x)$ is a good choice, and is used for example in vintage Yamaha FM synthesizers. It has the favorable property that over any constant length time interval, by the end of the interval the envelope has decayed to a constant fraction of what it was at the beginning of the interval. Damped oscillation (with some frequency-dependent energy loss) of a string is largely sinusoidal after enough time has passed, and if the oscillation is small enough that the system can be considered linear, then the amplitude loss per time unit will be proportional to the amplitude, giving an exponentially decaying envelope. Closer to the attack these assumptions do not hold.

In the simple exponential decay model, the choice of material could only manifest itself in the decay rate constant $b.$ If it would affect also $a,$ the player could adjust the plucking strength to compensate, nullifying this effect.

This will be very hand-wavy, but for small amplitudes (no non-linearities), the energy loss to the air depends on the acoustic impendance matching between the string, the instrument body, and the air. The acoustic impedance of air is $1.2 \text{ kg}/\text{m}^3 \times 344\text{ m}/\text{s},$ which is the product of the density and speed of sound. In solid materials both of these numbers are typically higher, and transmission of the vibration to the air, from a material of higher density and/or with a higher speed of sound, will require more reflections and will take more time compared to a material for which the figures are smaller. Here both the string and the instrument body are going to affect the outcome. The instrument body may also provide acoustic impedance conversion mechanically, so it's not all about materials.

Exponential decay of the amplitude is linear decay in dB scale.