I've found it depends largely on the organization and role you are pursuing. In my experience, larger corporate companies tend to have regimented qualifications that are fairly unwavering. Start-ups can be more flexible and have opportunities for on-the-job training.
In my own experience as a hiring manager, I've tended to ask for slightly more than is necessarily demanded of the position, which usually nets a more diverse array of applicants.
I find questions about problem solving can form the crux of a successful interview. This is an essential skill for any prospective employee and, conceptually, you can have experience with problem solving in your working life and beyond. Showcasing problem solving skills can compensate for a perceived lack of work experience.
I agree with Chris's input. Those are useful methods of getting in touch. I should add that LinkedIn is a great platform for incidentally making work connections and expanding your network. I've made contacts at other companies simply by realizing I had a friend or work acquaintance that knew someone at a company I wanted to reach out to. All it took was patience browsing LinkedIn profiles.
I'd also highly recommend checking out these companies on social media. A lot of recruiters will post job openings there. Beyond this, social is usually the forefront of the company's culture, so you'll get a good sense of whether they are truly a fit for you.