So, to recap:
The installation is correct and "/Lightstreamer" was the expected LS_HOME in the case shown.
The launch script output terminates correctly, because after that point any further log is to be expected on the file.
However, there is an initialization issue and the Server log is unsuitable.

Please modify conf/lightstreamer_log_conf.xml in the installation directory and change LightstreamerLogger.init to DEBUG for a clearer log.

Regarding the "//logs/LS.out" issue, this is strange, because the script was supposed to insert the LS_HOME, yielding somethig like /Lightstreamer/logs/LS.out
Do you still replicate the problem?