Learn Russian: Это уже десятая ошибка или только вторая? - Is this already the tenth mistake or only the second?

Ordinal numbers behave exactly like adjectives. They must match the gender, number, and case of the noun they describe. Since ошибка (mistake) is a feminine noun in the Nominative case (the subject of the sentence), we use the feminine forms десятая and вторая instead of the dictionary-form masculine десятый and второй.

Since ошибка is a feminine noun, why does the sentence start with the neuter word Это instead of Эта?

When you use это to mean 'This is...' or 'Is this...', it acts as a fixed pointing word and doesn't change gender, regardless of what follows it. If you instead wanted to say 'This mistake is the tenth', where 'this' attaches directly to the noun, then you would match the feminine gender: Эта ошибка — десятая.

Why isn't the word for 'mistake' repeated at the very end of the sentence?

Russian frequently drops nouns to avoid repetition, much like English does when saying 'only the second'. Because вторая has a feminine ending, it is grammatically obvious that it still refers to the feminine noun ошибка, making it unnecessary to say вторая ошибка.

How are уже and только being used here?