How to set [Bash on Ubuntu on Windows] [environment variables] from [windows path]?

I would try export JAVA_HOME="/mnt/c/Program Files (x86)/Java/jdk1.8.0_102" to set the JAVA_HOME variable in the bash shell.

Update (response to your edit):

I wouldn't recommend trying to automatically import your Windows paths to Bash on Ubuntu on Windows, because the paths have to be converted to be understood by the bash shell (\ to /, C:\ to mnt/c/ and so on), and because not all of the tools you're probably going to reference will work on both Windows and Linux. Instead, install what you need on the Bash shell using apt-get (you don't need to use sudo because BUW loads in a root shell). Java is probably fine to reference as above, but most things you'll want installed separately on Ubuntu.