@user61318 and @JosephORourke, thanks for the advertisement for my book.

@chubakueono, as far as I know the answer to your questions are no and no. Pages 148-149 in my book have the state of the art (essentially the formula you wrote, along with some additional background) and the relevant references. However, in a broader sense *much more* is understood about the asymptotic formula for $l_n$ and where it comes from. The main important points as relates to your question are:

The asymptotic formula for $l_n$ is closely related to a more detailed distributional limit for the random variable $L(\sigma_n)$ defined as the maximal length of an increasing subsequence in a uniformly random permutation $\sigma_n$ of order $n$. The result, known as the Baik-Deift-Johansson theorem, says that for any $t\in\mathbb{R}$,
$$ \mathbb{P}\left(
\frac{L(\sigma_n)-2\sqrt{n}}{n^{1/6}} \le t
\right) \to F_2(t) \quad \textrm{as }n\to\infty,
$$
where $F_2(t)$ is a certain probability distribution known as the Tracy-Widom distribution. $F_2(t)$ can be expressed explicitly as
$$F_2(t) = \exp\left(
-\int_x^\infty (x-t)q(x)^2\,dx
\right),
$$
where $q(x)$ is the unique solution to the Painleve II ODE
$$y''(x)=2y(x)^3+xy(x)$$
with certain asymptotic boundary conditions.

The constant $c$ in the asymptotic formula for $l_n=\mathbb{E}(L(\sigma_n))$ is simply the expected value of this distribution, that is
$$ c = \int_{-\infty}^{\infty} x F_2'(x)\,dx. $$

Proving a rate of convergence estimate in the asymptotic formula for $l_n$ would be extremely interesting (and, I suspect, difficult) and is an open problem as far as I know.

The Surprising Mathematics of Longest Increasing Subsequences, Cambridge University Press, 2015. (.) $\endgroup$ – Joseph O'Rourke Oct 11 '15 at 0:13Link to author page