Learn Russian: Подождите минуту, пожалуйста.

The command ending depends on the verb's stem. Because the core part of this verb ends in a consonant, we add -ите for polite or plural commands. If it ended in a vowel (like in читайте), we would use -йте.

Why is the word for minute spelled минуту here instead of минута?

It is in the accusative case. In Russian, when you state how long an action takes (the duration), you use the accusative case. Since минута is a feminine noun ending in -а, it changes to -у.

Could I just say ждите instead of подождите?

You could, but подождите sounds much softer and more natural for a polite request. The prefix по- gives the feeling of doing something "for a little while." Just saying ждите can sound like a strict, heavy order.

Why doesn't the spoken pronunciation of пожалуйста exactly match its spelling?

Because it is such a common everyday word, its pronunciation has been shortened over time. Native speakers almost always drop the й sound, making it sound more like "pa-zhal-sta".