Sound effects

It's almost always possible to add a one time sound effect. There is a limit based upon the current instrument definitions but you can define an instrument with a large number of options, my guess is about 127 sound fonts. If you use the violin, you can already have 3 sounds assigned to it for a C instrument. If you want a transposing instrument you can pick a trumpet and have 2 sounds defined. Just assign a sound to a channel in the mixer and the use staff text and set it to that channel. You don't have to assign a violin sound to the instrument or pizzicato sound to the violin's pizzicato channel. See https://musescore.org/en/handbook/mixer#mid-staff-change for how to change sounds mid staff.

https://soundfontdownloads.com/products/animals-sound-effects

If you want the sounds available for a single instrument then use a soundfont editor to make a mash-up. This way you can assign different effects to different pitches or velocities on the instrument. You may find that the soundfont linked above already does this.

I use Polyphone, (it's free)
https://www.polyphone-soundfonts.com/

I gave instructions in my link to how to change the sounds using https://musescore.org/en/handbook/mixer#mid-staff-change above. The only way to find free sound fonts for specific purposes is to search for them online.

You wrote:
...have a dog bark when a certain note is played, or a horn honk.

Effects like trumpet mute, pizzicato, harmonics, etc. are more related to music - and therefore supported in MuseScore - than are dog barks, horn honks, and (as someone once requested here in the forum) car engine sounds, etc.

An alternative...
If your sound effects are sparse and do not require specific pitch and meter notation, you can use readily available .wav or .mp3 sound effects. Then, use an audio editor application (like Audacity) to combine everything.
So... you export the MuseScore sheet as audio (.wav or .mp3), open in Audacity, and then add your audio sound effects within Audacity. This will produce an audio recording; but not a notated sheet (which a dog can't read anyway).
https://www.audacityteam.org/
(You can even record, then add, your own dog's bark :-)

Regards

For no clunkiness:

Add 1 extra instrument stave for all your sound effects and assign effects to different pitches in the soundfont. A piano would be a good option as it has a large range of pitches.

If you have any sound effects as .mp3 or .wav you can build them into a soundfont. Using Audacity is not necessary but it a really good idea since you can tidy up your samples, (e.g. normalise, remove noise, trim, reduce sample rate and therefore size), before combining them into a soundfont with Polyphone.

This will produce a notated sheet which plays back correctly, (maybe your dog can be trained to bark at the right moments). You will need to upload audio to ms.com, (just tick the box when uploading from Desktop), and downloaders will need your custom soundfont to hear it in MS desktop. You can share it via Dropbox, OneDrive, Google Drive, etc.

Re: Free sound effects, (albeit not ready made into a soundfont.)

Loads of them on YouTube, e.g. dogs @ https://youtu.be/iuy-oOJCOoM

Download the video and extract audio to make .wav samples to turn into a soundfont. Get other samples for other effects from other videos, (frogs, giraffes, helicopters etc.), and build them all into your soundfont. Add a piano stave to MS and assign it your effects soundfont.