Learn Russian: В городе много музыкальных театров. - There are many musical theatres in the city.

The word много (many / a lot of) always requires the noun phrase following it to be in the genitive case. Because we are talking about multiple theatres, it needs the genitive plural. The suffix -ых is the standard genitive plural ending for adjectives.

How do I know whether to use -ых or -их for an adjective like this?

You use -ых after standard hard consonants. You only switch to -их if the adjective stem ends in a soft consonant (like in синий) or one of the seven special spelling rule letters (г, к, х, ж, ч, ш, щ). Since the stem of музыкальный ends in a hard н, it takes -ых.

Why do the adjective and noun have completely different endings (-ых vs -ов)?

Adjectives and nouns have different sets of grammatical endings. While театров takes the standard masculine genitive plural noun ending -ов (from театр), the adjective describing it follows the adjective pattern, taking -ых. They perfectly match in case and number, even though their suffixes look different!

Why is В городе at the beginning, and where is the translation for 'there are'?

Russian doesn't use placeholder phrases like 'there is' or 'there are'. To express that something exists in a location, Russian simply puts the location first. So, 'In the city [are] many theatres' is the natural way to say 'There are many theatres in the city.'