Hyphenate all "self-"compounds, whether they are adjectives or nouns. Examples: "self-report technique," "self-esteem," "self-confidence."

Should self awareness be hyphenated?

Self-aware, with a hyphen. I wouldn't recommend checking Google when in doubt about a spelling. Check a dictionary. They're much more reliable.

When should two words be hyphenated?

Generally, you need the hyphen only if the two words are functioning together as an adjective before the noun they're describing. If the noun comes first, leave the hyphen out. This wall is load bearing. It's impossible to eat this cake because it is rock hard.