Learn Portuguese (Portugal): O senhor tem uma boa ideia?

Because the word for "idea", ideia, is a feminine noun. In Portuguese, the word for "a" or "an" changes to match the noun's gender, so we use the feminine uma instead of the masculine um.

Why does the translation say "Do you" when o senhor literally means "the sir" or "the gentleman"?

In Portugal, o senhor (for a man) and a senhora (for a woman) are used as the polite, formal way to say "you". Because it literally translates to "the sir", it acts like a "he" or "she" grammatically, so you must use the "he/she" form of the verb. That is why it uses tem (has) instead of an informal "you" verb form.

I thought adjectives usually go after the noun in Portuguese. Why is it uma boa ideia instead of uma ideia boa?

You are right that most descriptive adjectives go after the noun! However, a few very common, short adjectives like bom and boa (good) often come right before the noun.

How do you form a question like this without a word for "Do"?

Portuguese does not use helper words like "do" or "does" to ask questions. You just take the normal statement, keep the exact same word order, and add a question mark. When speaking, you simply raise the pitch of your voice at the end of the sentence.