This week's book giveaway is in the OCAJP 8 forum. We're giving away four copies of OCA Java SE 8 Programmer I Study Guide and have Edward Finegan & Robert Liguori on-line! See this thread for details.
Java class files have no concept of 32 vs. 64 bit - it makes no difference.
Classes compiled with Java 7 can run on earlier JVM versions provided appropriate -source and -target switches were used, AND the code doesn't use Java 7 features or doesn't use them on earlier JVMs. That's show it has always worked.