Grad School- What Are My Chances w/ No Gap Year?

Hi!
A little background info on me: I’m an undergraduate at Washington University in St. Louis with a major in Cognitive Neuroscience. I’m looking to get a Ph.D. in either clinical psychology or cognitive neuroscience. I’ll have a cumulative GPA somewhere in the 3.5s by the time I apply for graduate programs. I have no idea what my GRE scores will be like (haven’t started studying), but I tend to do fine on standardized tests. I assume my score won’t make or break my application, anyway. I’m primarily interested in researching the biological basis of mental disorders. I’ll have 4 semesters of research experience by the time I apply; 2 in a behavioral medicine lab and 2 in a cognitive neuroscience lab. I have professors/PIs who can write recommendation letters and I think they’ll be okay.
My biggest concern regarding graduate school is that I’m too mediocre to get in anywhere without a gap year. My parents aren’t the most supportive of my major/prospective career path, and they have made it very clear that a gap year is not an option for me. I’ve talked to them about how taking a gap year can make an application stronger, but they have told me that they will cut me off if I can’t get into a graduate school immediately (Fall 2017). (Their argument always boils down to something like this: “Why should we continue to help you out financially, and in doing so, imply that we have faith in you and your decisions, if a graduate program doesn’t think you can be successful at their institution?”) So that leads to a couple of things I was wondering if any of you guys could help me with:

  1. What caliber of schools/programs should I be applying to? I don’t have unlimited funds to apply to every program that exists, but due to the situation, I can’t afford to be too selective, wait for an opening with a particular faculty member (even if they’re really great fit), or aim too high than is realistic for me.
  2. If it’s becoming the norm to take a year (or a couple) off between undergrad and grad school, how competitive could I be if I’m applying with a junior spring GPA and less full-time research experience, publications, etc. than those who take a gap year?
    Thank you so much!