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I would try keywordtool.io - it's my go-to free tool for finding related keyword search terms. Simply plug-in a keyword into the tool and it will spit back a list categorized alphabetically. You are than able to copy the data into excel, and sort. Do this process a few times with various keyword variations and I'm sure you'll come across some good questions!

I've installed Google custom search engine (CSE) on multiple client websites. The $100 option is a good place to start, that gets you 20K queries.

Plus, over the past year Google has updated the features and UI allowing to easily customize the look and feel of the CSE. Attached is a screengrab showing the CSE UI.

Hope that helps!

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Yes, Google should allow you to use that landing page for you PPC Campaign. In fact you can use almost any landing page for a PPC campaign as long as it doesn't violate Trademarks and follows Google content policies.

The only drawback to using a landing page you don't own/control is you likely cannot embed remarketing or conversion tracking codes, which limits some of Adwords features.

Hope that helps!

I think your best bet is to view the "Paid vs Organic Traffic Report" in Google Adwords.

This report shows a comparison of when you're Organic and Paid traffic shows. I've attached a snapshot of how this report looks like in Google Adwords. And as Jasmine mentioned you'll need to connect Webmaster tools (or the search console as it's known now) to Adwords.

Hope that helps

GoF6DDr