Comparing currency prices If I know that one USD buys 0.90 euros or 1.42 Australian dollars, can I reliably conclude that one Australian dollar buys 0.63 euros? (0.90 / 1.42)

That seems to make sense. But would this be true for every case? For example, if you were to collect historic currency data for all currencies with respect to a single base currency, could you then accurately calculate all relative exchange rates, or are there are other factors to consider?
There can be market imperfections if information is not passed on immediately - historically that definitely happened all the time before telecommunication was well established.

Furthermore, if they have sufficient control of the in- and outflows countries can fix exchange rates for their currency at rates different from those of the free market. That happened in many socialist countries. The free market exchange rate for the Eastern and Western German currencies was substantially different from what the official exchange rate that was used by Eastern German banks. After unification of the countries again an exchange rate that was different from the market rate was used for the monetary union.

In Cuba a convertible peso was introduced in 1994 with a fixed exchange rate with the US dollar internally. It cannot be traded outside of Cuba.

The currency crises in Mexico and Asia in the 90s, and in Argentina in 2001-02 are associated with attempts to fix exchange rates that were in conflict with the market rates.