The variability in both clinical and counseling programs is so tremendous that there is almost no good answer to this question. Clinical psychology programs may be slightly more competitive, but this is changing and of course depends greatly on which programs you're talking about. Some counseling psych programs are more oriented to training academic counselors for college/university counseling centers. Some clinical psychology programs might be more focused on training psychotherapists than academic counselors. But one strong claim that can be safely made is that almost all clinical psychology PhD programs emphasize research more than clinical skills. This is also true of many counseling psychology PhD programs. The clinical PsyD degree, on the other hand, is focused almost exclusively on psychotherapy training. It makes little sense to apply only to clinical programs or only counseling programs; applicants should carefully study programs of both kinds and identify those that best fit their interests. Applicants with little interest in research would be well-advised to pursue a high-quality PsyD degree instead of a PhD.
What makes the distinction between clinical and counseling degrees really confusing is that there are plenty of counseling psychologists who decided not to work in academic settings, and so are employed alongside clinical psychologists in hospitals, mental health agencies and private practice. Of course, there are also a few clinical psychologists who work in university counseling centers; so as you can see, the distinction is blurry. One reliable difference, however, is that clinical psychology students are almost never trained in academic counseling.
There is almost no difference at all in the content of what you study. However, business schools tend to have more financial resources than psychology departments, and they also tend to pay their graduate assistants more. If you are interested in organizational psychology, it would be a good idea for you to read this web page about organizational behavior PhD programs.
It's been done before--by many, in fact. It's certainly difficult, but most single parents are well-acquainted with difficulty, I think. The effect it might have on your family is something you'll have to weigh out and decide for yourself.
Masters programs should take 2 years. Doctoral programs may take from 4 to 6, but 5 years is a good heuristic (that includes the Masters, so 5 years total). There are always a few students who take a long time to finish their dissertations, but these students are sometimes viewed unfavorably both by their departments and their potential employers unless they have legitimate excuses.
This can really vary. Masters students generally don't receive any aid because they're done quickly enough to find work and pay off their loans. Most of the financial aid packages I was offered in the PhD programs I was accepted into included tuition wavers for 4 out of the 5 years and enough money to pay for rent and groceries for a single student. Living stipends vary dramatically because cost of living varies so much. A tuition waver + a $12,000/yr stipend may sound like a lot until you remember that you'll be living in Los Angeles, for example. Also, if you're at a private school and get stuck paying your own tuition any particular year, that's easily $20,000+. Choose carefully.
In general, almost any kind of graduate school can be worth it as long as you don't amass huge amounts of debt along the way. Study after study shows that, on average, people who go to graduate school make slightly more money over the course of their lives than those who do not. If you're curious about how much money you'll be making if you go into the field, check out the results of this APA salary survey.