"Are there yellow flowers blooming in the yard?"
I think it should as I hear blooming as coming into bloom since flowers don't bloom, the plants that bear them do. The Japanese with に is used for what I'd render, "Are there yellow flowers having come into bloom in the yard?" Or more commonly, "Are there yellow flowers in bloom in the yard?"
No, "in" vs "at" is very much an English-specific thing where it depends on the nature of the space, and even varies from dialect to dialect. Kids can play both "in" and "at" the park (you'd use で for both in Japanese) and you can be "at" home (家に), and talk "in" Japanese (日本語で) etc. etc.
This is not a case of using the いる/ある "to exist" verbs. It is a verb conjugation indicating continuous action, and is created using the "て" form of a verb plus いる (or います to be formal). It doesn't matter that the flowers are inanimate because this sentence is talking about what the flowers are doing, not just that they're there.
さく (さきます) = to bloom
さいている (さいています) = to be blooming (continuous action)