First lets try to solve the missing LOG dir.
in your C:\Program Files\Elastic\Elasticsearch\6.6.1
add a new folder named LOG
Your elasticRun.bat (just to call it differently than the .exe) should look like this.
set JAVA_HOME=C:\Program Files\Java\jdk1.8.0_112
set Path=%PATH%;C:\Program Files\JAVA\jdk1.8.0_112\bin
This .bat should be in your elasticsearch\6.6.1\bin folder
now make your Elasticsearch.exe always run as administrator by Right-Clicking the Elasticsearch.exe and select Properties, then select the top tab named SECURITY and check the box that says "Run this Program as Administrator", press APPLY the OK to close box
now running the .bat file should work.
Make sure the JAVA dir I used in the bat is the correct ver to your system.
I tested this in my system Windows 10 x64 and worked fine.
IF and after you get the Elasticsearch to run.... to close it just press CTRL-C in the open elasticsearch window
Hope this works for you.
This will only work as a manually started Elasticsearch, since the Set commands in the bat files will not be applied system wide, just in the window opened by the .bat file.
To get ElasticSearch to start automatically as a service you will need to solve your PATH problem permanently. Or the service might not start...
But first it will be good to see Elasticsearch start at least manually.
If problems still occur, try removing ALL Previous java versions on PC, then installing JAVA SE JDK with the lastest version and update the .bat file with the new JAVA dir values if any changed.
Page to download the latest JAVA SE JDK is
Select the 64bit version (if your system is 64bit)
I'm using the Java SE Development Kit 8u202 windows x64 version.