Issues with RoboHelp Template

In the template editor there are many settings that you will have to experiment with and test thoroughly.  

Unable to reduce white space between panes. 

If I have followed correctly, what you are referring to as white space is panel width. Reduce that and less white space. You will likely need to increase one or more panels to compensate for a reduction in one.

left and right tab resize is not being responsive

I'm not sure what you mean here but is this setting applied?

Clicking Collapse icon on left is hiding text

Please clarify. The icon is meant to hide the left panel. See it on my site.

Even after collapsing output is not responsive

Again, please clarify.

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My site www.grainge.org includes many free Authoring and RoboHelp resources that may be of help.

I've edited your post to put the image into the page instead of as an attachment - that way is harder for people to see what you're describing.
I don't have any designer suggestions other than having a developer look at the coding. Perhaps you need to try out other designs to see if the features you're looking for are present in them - that may give some clues as to what to change in your template.
Not sure what you mean by "protect from unauthorized access of my help pages?" - RH doesn't do anything around access of the HTML5 it produces - that would have to be a discussion with your website designer and whatever tool they use to enable user access.