For the first problem, either use \tilde instead of \widetilde as egreg suggests or put an empty group before the letter:
\hat{\widetilde{{}A}}
though I would strongly suggest that you use \widehat with \widetilde. To illustrate
\begin{equation*}
\hat{\tilde{A}} \qquad
\hat{\widetilde{{}A}} \qquad
\widehat{\widetilde{{}A}}
\end{equation*}
produces

For the second problem, egreg's approach will give you the superscript at the height you desire. For the cases with \widetilde you can again either add an empty group before the superscript, or more generally a \vphantom:
\begin{equation*}
\hat{\tilde{A}}^{a} \qquad
\widehat{\widetilde{{}A}}{}^{a} \qquad
\widehat{\widetilde{{}A}}\vphantom{A}^{a}
\end{equation*}

All images produced from the following document. Note I have also loaded amsmath.
\documentclass[a4paper,11pt]{article}
\usepackage{amsthm,amsfonts,amsmath}
\begin{document}
\begin{equation*}
\hat{\tilde{A}} \qquad
\hat{\widetilde{{}A}} \qquad
\widehat{\widetilde{{}A}}
\end{equation*}
\begin{equation*}
\hat{\tilde{A}}^{a} \qquad
\widehat{\widetilde{{}A}}{}^{a} \qquad
\widehat{\widetilde{{}A}}\vphantom{A}^{a}
\end{equation*}
\end{document}
\widetildefor this;\tildeis the right option. – egreg Oct 23 '18 at 12:06latex,pdflatex,lualatex. – Henri Menke Oct 23 '18 at 12:26latexand frompdflatex. How did you produce your screen shots? – GuM Nov 01 '18 at 23:58