I want to build a 'Drawdio' Musical Pencil,(link to the make-project) but my local shop doesn't have the correct parts as originally used in this project. He doesn't have ceramic disc capacitors of 0.1 micro Farad, but he does have multilayer ceramic capacitors of the same capacitance. Can I use these capacitors or are the functionalities of both capacitors much too different?
Yes. Multi-layer ceramic SMD capacitors are generally a superset of the older disc capacitors. There is really nothing the old 100 nF disc capacitor can do that a modern 100 nF SMD cap can't do, within their voltage ratings.
Also, I'd look at the schematic carefully and see how that capacitor is used. If it's in the signal path, then it probably needs to be the value it is. If, however, it's just a bypass cap, you can safely replace it with a 1 µF ceramic SMD cap. 100 nF was used for decoupling back in the pleistocene, not because the value was optimal, but because that was the most capacitance you could get in a ceramic affordably. Today that tradeoff has moved on.
You don't want to go too far with increasing capacitance for a bypass cap because the larger ones, even multi-layer ceramic, have lower self-resonant frequencies and therefore higher impedance at really high frequencies. However, the 1 µF SMD ceramics of today have lower impedance across the whole frequency spectrum compared to the old thru-hole 100 nF disk capacitors of ancient times.
Almost certainly, Yes. Be aware that ceramic disc capacitors can be microphonic. That is: they can pick up vibration and introduce that vibration as a signal into your circuit.
Most ceramic capacitors do this to some extent but multi-layer ceramic caps are worse than disc capacitors.
Other than that, and so long as you meet the voltage rating, you can substitute one for the other.