Check out A Short, Comprehensive, Practical Guide to Copulas by Atillio Meucci. The paper provides further references in case you'd like to learn more. Steps #3 and #5 are addressed by this paper.
Step #4 is specific to this particular function which I am not too familiar. Since you have so many questions you'd probably have better success breaking down the problem into related parts. You will be in a better place to frame question #4 after some further background reading. As such you really can't answer the question without going into the background of copulas.
There are parametric and non-parametric methods to estimating copulas. In my view, parametric copulas impose tight restrictions that are not respected by the data (non-stationarity, fat tails, etc.). More recent research on time-varying copulas and Meucci's non-parametric copulas I believe can better cope with these issues.
How to sample from a given univariate CDF is a huge subject, so I will assume that part of the answer is known and will address how to find the conditional CDF from the copula.
By definition, any copula assigns probabilities to rectangular regions (within the unit square) delimited on the right by its first argument and above by its second argument. In particular, when $U$ and $V$ are uniformly distributed with $C$ as the copula for $(U,V)$ and $0 \lt \epsilon \le 1 - u$ is sufficiently small,
$$\eqalign{
\Pr(U\in (u, u+\epsilon]\text{ and }V \le v) &= \Pr(U\le u+\epsilon, V \le v) - \Pr(U\le u, V \le v) \\
&=C(u+\epsilon, v) - C(u, v).
}$$
Therefore, the conditional cumulative distribution function ought to arise as the (right-hand) limiting value of
$$\Pr(U\in (u, u+\epsilon]\text{ and }V \le v\,\Big|\,U\in (u, u+\epsilon]) = \frac{C(u+\epsilon, v) - C(u, v)}{\epsilon}.$$
Provided this limit exists (which it will almost everywhere for $u$), by definition it is the first partial derivative, $\partial C(u,v)/\partial u$. This, therefore, gives the conditional CDF for $V\,\Big|\, U=u$ evaluated at $v$.
The left figure shows a contour plot of the copula (representing a surface) $C(u,v)=uv/(u+v-uv)$. The right figure is the graph of the conditional distribution of $V$ for $u\approx 0.23$. It is a cross section of the rightward slope of the surface.
Reference
Roger B. Nelsen, An Introduction to Copulas, Second Edition. Springer 2006: Section 2.9, Random Variate Generation.
Best Answer
Your question is somewhat related as both are based on conditional probabilities and thus partial derivatives of copulas. However, the sampling cannot easily be obtained from the single density values, but from the conditional distribution function $F(y|x) = \frac{\partial}{\partial F_1(x)} C(F_1(x),F_2(y))$. For ease of notation, let us only look at the copula part. The strategy I would suggest is to use the general sampling scheme from a copula $C(u,v)$ (going back to the univariate probability integral transform) where one takes two independent random samples $\bf u$ and $\bf y$ of the same length from a uniform distribution $U(0,1)$, then: $$ {\bf v} = \left(\frac{\partial}{\partial u|_{u=\bf u}} C\right)^{-1}({\bf y})$$ In your case, you do not want to obtain a sample for $u$ and $v$ following $C$, but just a sample of $v$, given a particular value of $u$. Hence, you do not sample a vector $\bf u$, but just fix it at your desired value and sample a vector $\bf y$ that is used in the inversion of the partial derivative of the copula as in the general case above.
An approach in R could look like: