The ideas of Greek philosophers influence the world of today in many ways. Just a few of the more important would be scientific observation and classification, the question-and-answer method of teaching and learning, the moral advice that "just because it feels good doesn't mean one should do it", and the common-sense axiom that there is much more than what meets the eye in most any ordinary situation.
The ideas of Greek philosophers influenced our world by
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Greek civilisation was not influenced by Greek architecture. Greek architecture was a product of Greek civilisation
Rule of law
The thing you refer to as mythology was considered a religion. As much so as any today.
The ancient Greeks influenced modern government as, much like we do today, they had three main bodies of government. These were the Assembly, the Council, and Courts. Our US government today also has three branches; Executive, Legislative, and Judicial.
The Romans were a very innovative people, but one possible influence on their architecture would be the Etruscans, who are supposed to have invented the arch, beyond that most of what the Romans built was their own.