TL;DR : These parameters can indeed be controlled mechanically, though not to the extent enabled by today's synthesizers and software.
Pretty much all of the instruments of the traditional orchestra can adjust some or all of these parameters mechanically. They existed well before synthesizers were a thing, though the terminology was different.
Let's take the piano as an example:
Frequency (pitch) is determined by the length and diameter of the string, and controlled by the keys.
As for volume adjustment, the piano can play a range of dynamics, from quiet (pianissimo) to loud (fortissimo). That's actually the killer feature that made piano so popular during the Romantic period. Mechanically, this is caused by hammers hitting the strings with varying strength.
Filtering occurs in conjunction with dynamics. The higher the volume, the brighter the sound (meaning more energy in the high part of the spectrum). The soft pedal can also be used as a "low-pass-ish" filter.
When it comes to synthesis in the mechanical world, the pipe organ can arguably be considered the very first synthesizer in Western music. By combining sets of pipes, it is possible to change the overall sound, much like performing additive synthesis.
Edit: As pointed out in the comments, ADSR envelope can be controlled mechanically to some extent, for example using the sustain pedal of the piano. The envelope of string instruments can be modified by using - or not using - a bow: think of the double bass in classical music (bowed) VS double bass in jazz (fingered).
It is worth noting that, in purely mechanical instruments (and arguably in analog synthesizers), it is almost impossible to adjust one parameter without another varying accordingly, for example frequency and volume, or volume and filtering. Think of an oboe player trying to produce the highest notes of the instrument at low volume, that's near impossible.
As for arpeggiator, you might want to take a look at punched cards and how they were used to generate melodies, typically played by barrel organs. This contributed to the start of electronic music at the beginning of the 20th century.
4It may be worth researching "foley" and "old time radio" and even "cartoon sounds" (I was listening to a 20khz/99%Invisible podcast about cartoon sounds recently). – April Salutes Monica C. – 2019-04-23T13:08:01.840
Your title asks about sounds, but your checklist asks primarily about flexibility and functionality. There are many mechanical tricks that can help with the latter, but not as much with the former. – Alexander – 2019-04-23T16:32:36.343
6I mean, ultimately all "electronic" sounds are generated mechanically, by means of physically vibrating discs in your speakers. – Darrel Hoffman – 2019-04-23T16:35:52.960
I'm a bit confused about potential conflicts here. Are you talking about adding these effects to an input signal on the fly, such as being able to change the volume as you listen to this mechanical playback? Or are you saying that with a given input signal, can you make a mechanism that recreates that sound but with modifications (such as volume/pitch/anything); but must be made in advance? Your final paragraph suggests the latter to me; but the rest of the question suggests the former. – JMac – 2019-04-23T18:34:33.477
FWIW, in principle you can even do human voice this way. – R.. GitHub STOP HELPING ICE – 2019-04-23T19:42:07.643
2Not only are electronic sounds played using mechanical means like a vibrating disc as mentioned by Darrel Hoffman but a lot of those sounds are originally sourced by real world sounds that are then recorded and changed according to the end goal. – Muuski – 2019-04-23T23:17:22.363