It is more likely true. Main reason is that you have more control over what your ad looks like and says then your organic listing. The only data I have to support is one client to which I had access to Analytics and had a top of first page organic listing for the same keywords. My ads would get higher conversion rates. Of course, the ads themselves being better may have something to do with it. But look at results when doing searches. Ads are cleaner which makes people be attracted to them and are easier to read. This makes people more inclined to click them and thus to buy.
Hadn't heard this before but it definitely makes sense. The fact that the ad looks exactly how you want makes a huge difference.
Visitors will convert if your website has relevancy to them. If you targeted the right group of people (country, language) and keywords, then your content interests them to some degree. So yes, they are more likely to convert.
The danger behind PPC campaigns is the fact that you are putting your paid ad in front of your organic listing. For many customers, research is an integral part of the process and they may go from site to site on the SERPs before they convert. If your organic listing shows up on page 1 anyways, the ad would be a waste of money. At least in my case, my organic traffic fell as my PPC traffic nearly doubled.
Well, it depends how relevant the ad is to what you're looking for. If someone is searching "buy home insurance" and come to your site about home insurance...then probably. If you're being more vague in your ads, it's not as likely.
If someone searches for "buy home insurance" they will most likely research more than the PPC ads, and if your website ranks in the first page, this would translate into wasted ad spending since the customer would eventually reach your site. I'm assuming we all understand how to write relevant ads. What's more important, when we toss each other advise, are the products and services that we're selling. For specific, identical products, such as electronics and cars, ad position (and Page Rank) may matter. For more particular products in which the customer would be more inclined to conduct research, ranking may be negligible if you rank organically.