[...] 6_u37 was released by Apple with the Developer Package release in October of 2012. I wouldn't call that 'pretty old'.

At an age when Apple and its fanboys overexagerrate any security hole in Java because Apple took their time to fix a known hole a year ago, that revealed to be used in the very first virus under Mac OS X, it's an eternity!

OS X will install 6_u43

See, you already missed one. Java 6 release version is now 6_u45! [;)]

Can you recommend running it w/ Java 7 on OS X?

Don't try Java 7 with Sweet Home 3D yet, it's still too buggy as I reported at the end of the blog article about version 4. Seriously, Oracle released a Mac OS X version of the JDK too early.
But since you seem to be able to use the Terminal, please try this:
- download Jar executable version of Sweet Home 3D
- run the program with the command: In case, you didn't know, you can get easily the full path of the SweetHome3D-4.0.jar file with a drag and drop of that file in the Terminal window.
- if it still fails, some Java log starting by "Exception in thread" should appear in the Terminal window. Please report it here.

Thanks for your patience.