Learn Russian: Откройте окно, пожалуйста.

Откройте is the formal or plural command form. To make it, we look at the "I" form of the verb открыть (to open), which is я открою. We drop the -ю to get the stem откро-. Because this stem ends in a vowel, we add -й for the informal command (открой) and -йте for the formal or plural command (откройте).

What case is окно in here, and why didn't the ending change?

It is in the accusative case because it is the direct object (the thing being opened). For neuter nouns like окно, the accusative case looks exactly the same as the dictionary (nominative) form. That is why it doesn't change to an -у or -ю like feminine nouns do.

How is пожалуйста actually pronounced? Is the й silent?

Yes! Even though it is spelled пожалуйста, native speakers almost always drop the й and the у when speaking at a normal speed. It usually sounds like "пожалста" (pa-ZHAL-sta). You still must spell it with all the letters, though!

Is the comma before пожалуйста required?

Yes. In Russian punctuation, the word пожалуйста is always set off by commas. If it comes at the end of a sentence like it does here, you must put a comma right before it.