Some people call this type of construction a secondary predicate. Secondary predication is sometimes divided into two types:
- Resultative: I punched him silly.
- Depictive: I drove home drunk.
In both examples, the secondary predicate describes the subject. In the former, it describes the result of the primary predicate--in other words, the state the subject is in after the primary predicate is complete. In the latter, it depicts the current state of the subject.
To put it another way:
- Resultative: state of the subject after the primary predication, as a result
- Depictive: state of the subject during the primary predication
In both of your examples, the secondary predication appears to be depictive, meaning it describes the state of the subject:
"He was standing there proud." → "He was standing there, being proud."
"He is growing up rich." → "He's being raised in a rich family."
Are they common? Relatively. Secondary predication of both types is productive in English, which means people use it to create new sentences with some regularity.
Unfortunately, there's usually no way to tell which interpretation is intended from the grammar alone. You have to tell from context whether you're looking at a resultative or depictive construction, and in some sentences both interpretations are possible.