Just wondering how search engines particularly Google treats links not in the <a> element. Do they count them as the links in the <a> element or as inbound/backlinks?
They count the same as long as they are do follow. But the anchor text is your URL so you better have your keyword in your domain name or it is not going to be very valuable.
I would think Google does count them as links, But I don't think search engines add any benefit to the page linked to.
Even if a link is nofollow, you can find those links in Google WMT backlinks. But by using only url of a site (without hyperlink ), search engines not consider them as backlinks but helps to improve site branding.
Right it depends on the on page SEO of the site with the link. Is your keyword in their domain or URL? Is it in the title? Is it located next to the http link? If it is then it is just as good as a keyword as the anchor. If it is an apparently unrelated to your topic page and you do not have the keyword in your domain name then the value of the link is greatly diluted.
Links that aren't in an anchor element? All hyperlinks have to be in the anchor element otherwise they're not links.
If you want to work for a particular keywords and want to rank high that keyword, you must create hyperlink on that keyword, then only that particular keyword will display in SERP. Otherwise only site link will not give any weightage to any of your keyword. But we can't say the site links are not important at all. As other member said it will increase site's brand value.