I almost replied yesterday about the 32 & 64 bit java but decided to install LW on Windows 7 first. From memory this is a 64 & 32 bit Java issue. I would always recommend downloading the Offline version, which is usually the 2nd option & is usually around 15 MB. Strangely on mine, LW seems to run fine with Java 32 bit on Win 7. Having more than one Java version may cause an issue however. Let us know how you go with trying the other Java version.
You can find out which Java version you have via the Java Control Panel. Also, in this Control Panel is a Security tab, check Java is set to allow security certificates. This might be your issue at least with downloading the Online Java since Java site does require this option AFAIK.