Should I try to make money off of my dollars?

Well, it depends on the exchange rate at that point in time. Talk to your banker and definitely you'll get a better advice. Also, do well to study the exchange rate on the market and you can change it when the Dollar$ rate is lowest, with that the other currency you want it changed to will be higher and you won't suffer any losses. Good luck.

Good question!

This depends on how liquid are you.  Hopefully not more than 3 months salary, as there are opportunities elsewhere. 

If it is less than 3 months salary, best to keep it liquid as possible, and the amount wouldn't be able to be leveraged for large sums from marginal gains.  So follow on if it is more than 3 months salary, and only use the excess of the 3 months salary.

What we are assuming though are two parts:  inflation will wipe out some of the value, which you want to hold, and there are opportunity costs to keeping money purely liquid.

Based on that, yes invest the excess elsewhere.  As to focusing on other currencies than the US dollar, there is another thought to this.  Why focus on foreign currencies as the only investment instrument?  If it is a concern of keeping everything quite liquid still, then yes, go for another currency.  If it is looking for an investment opportunity, then please look at other investment instruments (stocks, bonds, futures, etc).

Best of luck