How to change arrowhead size and scale in TikZ?

Question

I have been working on this figure and I have some problems.

1. How can I change the size of the arrowheads?
2. How do I place zero directly below the vertical line?

3. How do I change the scale of the figure?

   \documentclass[a4paper,oneside,article]{memoir}
   \usepackage[utf8]{inputenc}
   \usepackage[danish]{babel}
   \usepackage{babel}
   \usepackage{pgf,tikz}
   \usepackage{tikzscale}
   \usetikzlibrary{angles,calc,intersections,quotes,arrows}
   \usepackage{tkz-euclide}
   \usetkzobj{all}
   \usepackage{mathrsfs}
   \usetikzlibrary{babel}
   \begin{document}
   
   \begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
   \draw[<->,color=black] (-1.5,0.) -- (1.5,0.);
   \foreach \x in {-1.,1.}
   \draw[shift={(\x,0)},color=black] (0pt,2pt) -- (0pt,-2pt) node[below] {\footnotesize $\x$};
   \draw[->,color=black] (0.,0) -- (0.,3.5);
   
   \draw[color=black] (0pt,-10pt) node[right] {\footnotesize $0$};
   \clip(-1.5,-1.5) rectangle (6.5,3.5);
   \draw (-1.,3.14159)-- (-1.,0.);
   \draw (-1.,0.)-- (1.,0.);
   \draw (1.,0.)-- (1.,3.14159);
   \draw (-1.,3.14159)-- (1.,3.14159);
   \draw (2.05,-0.95)-- (5.92,2.92);
   \draw (2.34,1.7)-- (4.98,0.17);
   \draw(3.5,1.5) circle (1.cm);
   \draw [<->] (3.5,1.5) -- (3.933479957929779,2.401163207234515);
   \draw [<->] (1.3,1.5) -- (2.2,1.5);
   \begin{scriptsize}
   \draw [fill=black] (0.5,0.7854) circle (1.5pt);
   \draw[color=black] (0.48297257459254965,0.96005462744924652) node {$(p_2,\alpha _2)$};
   \draw [fill=black] (-0.3,2.6179) circle (1.5pt);
   
   \draw[color=black] (-0.490091064358716,2.8276333241269166) node {$(p_1,\alpha _1)$};
   
   \draw[color=black] (-1.3379064613731417,3.14159) node {$\pi$};
   \draw[color=black] (5.8,2) node {$T(p_2,\alpha _2)$};
   \draw[color=black] (5.2,0.4451160520474517) node {$T(p_1,\alpha _1)$};
   \draw [fill=black] (3.5,1.5) circle (1.5pt);
   \draw[color=black] (3.6514466357055486,2) node {$1$};
   \draw[color=black] (2.6676305320562292,2.611406719382492) node {$b(o,1)$};
   \draw[color=black] (3.3163103462895014,1.5897712463899266) node {$o$};
   \end{scriptsize}
   \end{tikzpicture}
   \end{document}
   

Answer 1

You can change the size of the arrowheads after loading the newer arrows.meta library using, for example, >={Stealth[inset=0pt,length=8pt,angle'=28,round]}. Here you can change the type of arrow tip and its length/inset/angle/round/etc.

To place the zero directly below the vertical line, do \draw (0pt,-2pt) node[below] {\footnotesize $0$};, where (0pt,-2pt) is at the same level as the 1 and -1 to its right and left, respectively.

To change the scale of the figure you can simple say scale=<ratio> instead of x=<> and y=<>. Also, there is xscale=<> and yscale=<> to have separate scaling horizontally or vertically.

Last, remove all draw=black and instead of fill=black just use fill, black is the default.

\documentclass[a4paper,oneside,article]{memoir}
\usepackage[utf8]{inputenc}
\usepackage[danish]{babel}
\usepackage{babel}
\usepackage{pgf,tikz}
\usepackage{tikzscale}
\usetikzlibrary{angles,calc,intersections,quotes,arrows.meta}
\usepackage{tkz-euclide}
\usetkzobj{all}
\usepackage{mathrsfs}
\usetikzlibrary{babel}
\begin{document}

\begin{tikzpicture}[line cap=round,line join=round,>={Stealth[inset=0pt,length=8pt,angle'=28,round]},scale=1.5]
\draw[<->] (-1.5,0.) -- (1.5,0.);
\foreach \x in {-1.,1.}
\draw[shift={(\x,0)}] (0pt,2pt) -- (0pt,-2pt) node[below] {\footnotesize $\x$};
\draw[->] (0.,0) -- (0.,3.5);

\draw (0pt,-2pt) node[below] {\footnotesize $0$};
\clip (-1.5,-1.5) rectangle (6.5,3.5);
\draw (-1.,3.14159)-- (-1.,0.);
\draw (-1.,0.)-- (1.,0.);
\draw (1.,0.)-- (1.,3.14159);
\draw (-1.,3.14159)-- (1.,3.14159);
\draw (2.05,-0.95)-- (5.92,2.92);
\draw (2.34,1.7)-- (4.98,0.17);
\draw(3.5,1.5) circle (1.cm);
\draw [<->] (3.5,1.5) -- (3.933479957929779,2.401163207234515);
\draw [<->] (1.3,1.5) -- (2.2,1.5);
\begin{scriptsize}
\draw [fill] (0.5,0.7854) circle (1.5pt);
\draw (0.48297257459254965,0.96005462744924652) node {$(p_2,\alpha _2)$};
\draw [fill] (-0.3,2.6179) circle (1.5pt);

\draw (-0.490091064358716,2.8276333241269166) node {$(p_1,\alpha _1)$};

\draw (-1.3379064613731417,3.14159) node {$\pi$};
\draw (5.8,2) node {$T(p_2,\alpha _2)$};
\draw (5.2,0.4451160520474517) node {$T(p_1,\alpha _1)$};
\draw [fill] (3.5,1.5) circle (1.5pt);
\draw (3.6514466357055486,2) node {$1$};
\draw (2.6676305320562292,2.611406719382492) node {$b(o,1)$};
\draw (3.3163103462895014,1.5897712463899266) node {$o$};
\end{scriptsize}
\end{tikzpicture}
\end{document}