Learn Farsi: سفیر جدید خیلی باادب و باحوصله است. - The new ambassador is very polite and patient.

The prefix با means "with". It turns a noun into an adjective describing someone who has that quality. Here, ادب means "manners", so باادب means "polite" (with manners). Similarly, حوصله means "patience", so باحوصله means "patient" (with patience).

How would I form the opposites of these two adjectives?

You can use the prefix بی which means "without". By replacing the prefix, you get their exact opposites: بی‌ادب means "impolite" or "rude" (without manners), and بی‌حوصله means "impatient" or "bored" (without patience).

Is there a hidden Ezafe connecting the first two words?

Yes, there is an Ezafe sound connecting the noun to its adjective. You pronounce it as safir-e jadid. The Ezafe is the short "e" sound that links them together, though it is not written in the Persian alphabet.

Why are باادب and باحوصله written as single connected words?

When با is used as an adjective-forming prefix, it is typically attached directly to the noun in writing, or connected with a half-space. If با was being used as the preposition "with" in a sentence, it would be written as a completely separate word with a full space.