You are probably looking for the delta method, see for example:
Oehlert, G. W. 1992. A note on the delta method. American Statistician 46: 27–29.
There maybe more to it, but to me it seems that you just want to determine goodness-of-fit (GoF) for a function f(a), fitted to a particular data set (a, f(a)). So, the following only answers your third sub-question (I don't think the first and second are directly relevant to the third one).
Usually, GoF can be determined parametrically (if you know the distribution's function parameters) or non-parametrically (if you don't know them). While you may be able to figure out parameters for the function, as it appears to be exponential or gamma/Weibull (assuming that data is continuous). Nevertheless, I will proceed, as if you didn't know the parameters. In this case, it's a two-step process. First, you need to determine distribution parameters for your data set. Second, you perform a GoF test for the defined distribution. To avoid repeating myself, at this point I will refer you to my earlier answer to a related question, which contains some helpful details. Obviously, this answer can easily be applied to distributions, other than the one mentioned within.
In addition to GoF tests, mentioned there, you may consider another test - chi-square GoF test. Unlike K-S and A-D tests, which are applicable only to continuous distributions, chi-square GoF test is applicable to both discrete and continuous ones. Chi-square GoF test can be performed in R by using one of several packages: stats built-in package (function chisq.test()) and vcd package (function goodfit() - for discrete data only). More details are available in this document.
Best Answer
I am not terribly familiar with R but I believe the standard way to perform nonlinear regression is using the
nlsfunction. Since you do not say what specific model you are trying to fit to the data, I cannot help you any further. But maybe this small tutorial will help.Regarding the adequacy of the model, R-squared is indeed not a good statistic. Maybe you can try to compare two or more models using the AIC or BIC values.