Were there established "denominations" for coins of different metals in ancient times?

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Yes, denominational coinage, by definition, has a set relationship between the units and that relationship will not be exactly the same as the relative value of the metals involved. For example, the Roman denarius was originally denominated to be worth 10 asses. That's why it is called a "denarius" which means thing of ten.

This does normally cause a problem because the value of a coin is usually above its metal value because there is a value in the standardization and the authority of the issuer.

If, however, this seigniorage value is insufficient, then people will start melting the coins down, so in that case either the denominations need to be changed or the coinage needs to be debased (or upgraded). So, for example, the denarius was re-denominated to be worth 16 asses in 211 BC.

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