Learn Russian: Где мышь? - Where is the mouse?

Nouns ending in a soft sign (ь) can be either masculine or feminine, so you often just have to memorize their gender. However, мышь (mouse) is feminine, and there is a handy rule here: any noun ending in a hissing letter (ж, ш, ч, or щ) followed by a soft sign is always feminine!

If the letter ш is always a hard sound, does the soft sign actually change how мышь is pronounced?

No, it doesn't! You are right that ш is always hard, so the soft sign (ь) is completely silent here. In this word, the soft sign is just a spelling rule acting as a grammatical marker to show that the noun is feminine.

Where is the word for "is" in this sentence?

In the present tense, Russian drops the verb "to be" (am, is, are). You don't need a word for it at all. So instead of saying "Where is the mouse?", you literally just say "Where mouse?" (Где мышь?).

Does мышь only refer to the animal, or can it be a computer mouse too?

It means both! Just like in English, you can use мышь for the animal and for the device you use with your computer. You will also often hear Russians call a computer mouse a мышка, which is a cute, affectionate version of the word.