Can you say that confidence intervals measure worst case and best case? Suppose that I make a point estimate, 0.7, with a 90% CI: [0.6, 0.8].
Can I say that in the worst case, the true parameter is 0.6 and in the best case it's 0.8?
 A: The true parameter value simply is whatever it is.  It isn't clear what "best case" or "worst case" could mean.  You might be happy or sad about the actual value of the parameter, if you could magically find out its true value, but it is constant.  
Both @PeterFlom and @R.Carlos have accurately explained what a confidence interval is.  Here is another way to think about it.  In your case, if you had chosen $.6$ (or $.8$) as your null hypothesis and conducted your test at the $\alpha=.10$ level, you would not have rejected the null, but if you had chosen $.59$ (or $.81$) instead (or tested at the $\alpha=.05$ level), you would have rejected the null.  
A: As Peter said. Reapeating your experiment x times and calculating CIs everytime then 95% of these CIs will contain the real population parameter which you tried to estimate. The parameter is either in or not. There is no probability assigned of how close your estimate is to the population parameter. 
A: No.  That is not a correct statement; it might not even be a meaningful one.
The Wikipedia page for confidence intervals is pretty good, I think.  In particular, this line:

When we say, "we are 99% confident that the true value of the
  parameter is in our confidence interval", we express that 99% of the
  hypothetically observed confidence intervals will hold the true value
  of the parameter.

is a good explanation of what CIs really are. 
