Eurodollar Futures - How to interpret the price and correlation between EUR and USD?

In general the eurodollar future is an efficient way to gain or reduce risk to short term USD interest rates. Even though it is a very liquid futures contract, it has almost no effect on the cash FX market (spot or forward), as the later is much, much bigger. If you want to understand the term structure of the FX market, look at FX forward rates.

A rise or fall of the eurodollar futures price can't tell you anything about what the USD FX rates will do anyway, because the information therein is already part of the cash FX forward price. The term structure can only tell you what the cash price will be in the future assuming that the market does nothing else (which of course never happens).