Calling it a text adventure, although it also has graphics, is what made the confusion. Especially as it has been a lot of talk about making an engine for text-only adventure games on the forums recently.
Krzysiobal made an MMC5 clone
repro. But it looks like he didn't clone the audio chip which is a big point in this game according to Mr Yoshimiru.
That "timer" is for controlling the frequency of the sound channel. It is not the duration of the sound. It used to be called "period" in the wiki but it seems to have been changed to "timer". Probably technically correct but confusing. The length counter is what controls the duration of the sound.
MMC5 pulse channels are pretty much the same as the APU's pulse channels, except they have no sweep hardware. BTW a sweep is changing frequency over time and an envelope is changing volume over time.
APU pulse 1
$4000 envelope/volume and duty cycle
$4001 sweep
$4002 frequency low
$4003 length and frequency high
APU pulse 2
$4004 envelope/volume and duty cycle
$4005 sweep
$4006 frequency low
$4007 length and frequency high
MMC5 pulse 1
$5000 envelope/volume and duty cycle
$5001 (unused)
$5002 frequency low
$5003 length and frequency high
MMC5 pulse 2
$5004 envelope/volume and duty cycle
$5005 (unused)
$5006 frequency low
$5007 length and frequency high
Keep in mind that, like Dwedit said, sound engines often don't use the hardware length counter, envelope or sweep (they often disable them). Instead they do these things manually in software. Duration is done by unmuting and muting the channel with the desired timing, envelope is done by manipulating volume every frame as desired and sweeping is the same but by manipulating frequency instead.
For that reason, looking for writes to the sound registers might not do you any good. You need to figure out what byte or bytes are controlling the duration of the sound effect in the sound effect data that is loaded.
Edit: If $5004 gets set to $30, the hardware envelope and length counters are both disabled. So the game do indeed use software envelope and duration.