Modified Erdos-Renyi Random Graph with Self-Loops and Maximum Allowed Degree of Nodes

Question

Let $G(n, p, k)\ (n, k \in \mathbb{N}^+, p \in \mathbb{R}, 0<p<1)$ be a graph with $n$ nodes constructed with the following rules:

1. The degree of any node does not exceed $k$.
2. Each edge is included in the graph randomly, with an expected number of edges equal to $npk/2$, provided that rule 1 is not violated. Self-loops are allowed.

Let $c_i$ be the fraction of nodes in a connected component with $i$ nodes. Here are the properties of the graph that I would like to know when $n\rightarrow\infty$:

1. For any $k$, is there a critical value of $p$ where the expected number of nodes in each connected component, $\mathbb{E}(n_c)=\sum_{i=1}^{\infty}ic_i$ reaches infinity?
2. Is it possible to express the distribution of $c_i$ as a function of $k$ and $p$?
3. Instead of restricting the maximum degree of nodes to $k$, if we consider each node to have $k$ unique "connection points" and edges may be constructed between any "connection points", will the answer to the two previous questions be different?

I'm not a math major so the question might not be worded in the most accurate or rigorous way, but nonetheless I would appreciate any help on this. Thanks!