Does Google Rank A Website Better If It Is Using Adwords?

Yes it would help you. Adwords is a service by google and if you use their service you would definitely get good rank. On the other hand, Google is the most successful IT Solution Provider in the world today. You should go for Adwords.

Actually, No.  AdWords is simply the service that Google has to allow you to advertise your site (product / service).  It has nothing to do with the page ranking that Google assigns.

The ranking of your page in the search results has to do with search engine optimization (SEO) and obtaining solid backlinks to your site.

Google Adwords may get you site shown in a "relevant" search that someone makes, but when they click on the link, you are charged according to the amount that you have bid to have your site be shown on that particular search.

I suggest researching SEO and optimizing your content, obtain some backlinks and get ranked through hard work.

No, it is so simple. You can check by using google position tool and see if a websites rank changes just after they adwords.