Learn Russian: В спальне нет табуретки.

The word нет meaning "there is no" always requires the following noun to be in the Genitive case to show absence. For the feminine noun табуретка, the -а changes to -и because Russian spelling rules do not allow the letter ы after к.

Why does в спальне end in -е? Doesn't the word for bedroom end in a soft sign?

It is easy to mix this up, but the dictionary form of "bedroom" is спальня, which ends in the vowel -я. Feminine nouns ending in -я follow the standard rule for the Prepositional case and take an -е ending. Only feminine nouns ending in a soft sign, like кровать (bed), take an -и ending to become на кровати.

I have seen the word табурет for stool. Is табуретка different?

They mean the exact same thing, but табуретка is feminine and is much more commonly used in everyday spoken Russian. Табурет is masculine and sounds a bit more formal or technical, like something you would see in a furniture catalog.

If I wanted to say "There are no stools" (plural), would the sentence change?