Arrows inside a commutative diagram using tikzcd

You can shorten a Rightarrow as you like.

For the triangle, you can name the label and draw an arrow to it.

\documentclass[12pt,reqno,a4paper]{amsart}
\usepackage{tikz-cd}
\tikzcdset{row sep/normal=50pt, column sep/normal=50pt}

\begin{document}

\begin{tikzcd}
    A\arrow[d, "p"']\arrow[Rightarrow, shorten >=25pt, shorten <=25pt, dr]\arrow[r, "q"] &  B\arrow[d, "r"]\\
    C\arrow[r, "s", swap] & D
\end{tikzcd}

\begin{tikzcd}
    A\arrow[d, "p"']\arrow[r, "q"] &  B\arrow[d, "r"]\\
    C\arrow[r, "s", swap]\arrow[Rightarrow, shorten >=25pt, shorten <=25pt, ur] & D
\end{tikzcd}

\begin{tikzcd}
    A\arrow[d, "p"']\arrow[rd, "q"{name=M}] \\
    B\arrow[r, "r", swap]\arrow[Rightarrow, shorten >=10pt, shorten <=10pt, to=M] & C
\end{tikzcd}

\end{document}

Why not simply this, to specify the diagram commutes? Unrelated: I took the liberty to simplify your code, in particular removing unnecessary packages (already loaded by another package).

\documentclass[12pt,reqno,a4paper]{amsart}
\usepackage{extsizes}
\usepackage{blindtext}
\textheight 9.3in \textwidth 6.5in
\calclayout
\usepackage{mathtools}
\usepackage{amsthm,amssymb}
\usepackage{hyperref}
\usepackage{mathrsfs}
\usepackage[all]{xy}
\usepackage[normalem]{ulem}
\usepackage{tikz-cd}
\usepackage[utf8]{inputenc}
\usepackage[english]{babel}
\pagestyle{myheadings}

\begin{document}

\begin{tikzcd}[row sep = large, column sep = large]
 A \arrow[d, "p"'] \arrow[r, "q"] & B \arrow[d, "r"] \\
 C \arrow[ur, phantom, "\scalebox{1.5}{$\circlearrowleft$}" description]\arrow[r, "s"] & D
 \end{tikzcd}

 \end{document} 

You can use a phantom arrow, with the label in the middle via description and sloped.

\documentclass[12pt,reqno,a4paper]{amsart}
\usepackage{tikz-cd}

\begin{document}

\begin{tikzcd}[sep=huge]
A \arrow[dr,phantom,"\to" description, sloped]
  \arrow[d, "p"'] \arrow[r, "q"] & B \arrow[d, "r"] \\
C \arrow[r, "s"] & D
\end{tikzcd}

\end{document}