The problem you have has two parts.
For the first part (text overlapping in right margin) please have a look to my answer to question. There you find the explaination for the in the following MWE used code:
\newcommand*{\mycventry}[7][.25em]{
\begin{tabular*}{\linewidth}{l@{\extracolsep{\fill}}r}% <=============
{\bfseries #4} & {\bfseries #5}\\%
{\itshape #3\ifthenelse{\equal{#6}{}}{}{, #6}} & {\itshape #2}\\%
\end{tabular*}%
\ifx&%
\else{\\%
\begin{minipage}{\linewidth}% <=====================================
\small#7%
\end{minipage}}\fi%
\par\addvspace{#1}}
For the second part, the possition of the bullet, you can use the line
\renewcommand\labelitemi{\raisebox{0.8em}{\oldlbi}}
to correct the possition of the bullet. Because you do not know what is comming after that code it is important to use
\renewcommand\labelitemi{\raisebox{.15em}{\oldlbi}}
afterwards to get the standard position back.
So the best would be to create a new command \mycventrya
:
\newcommand{\mycventrya}[4]{%
\renewcommand\labelitemi{\raisebox{0.8em}{\oldlbi}}
\begin{itemize}
\item{%
\mycventry{#1}{}{#2}{#3}{}{}
\vspace{-1.5\baselineskip}
\begin{changemargin}{0cm}{2.5cm}
#4
\end{changemargin}
}
\end{itemize}
\renewcommand\labelitemi{\raisebox{.15em}{\oldlbi}}
}
Now you can add your informations with the line:
\mycventrya{1963--1967}{PhD Dolphinology}{University of Atlantis}{Over the course of my research I became fluent in dolphin communication, developing a transcription method which I subsequently used in the writing of my thesis.}
With the following MWE
\documentclass[11pt,a4paper,sans]{moderncv}
\moderncvstyle{banking}
\moderncvcolor{blue}
\usepackage[utf8]{inputenc}
\usepackage[scale=0.8]{geometry}
\usepackage{import}
% from https://tex.stackexchange.com/questions/588/how-can-i-change-the-margins-for-only-part-of-the-text
% change margins on the fly
\def\changemargin#1#2{\list{}{\rightmargin#2\leftmargin#1}\item[]}
\let\endchangemargin=\endlist
% https://tex.stackexchange.com/questions/392811/moderncv-extend-hrule/392894#392894
\newcommand*{\mycventry}[7][.25em]{
\begin{tabular*}{\linewidth}{l@{\extracolsep{\fill}}r}% <=============
{\bfseries #4} & {\bfseries #5}\\%
{\itshape #3\ifthenelse{\equal{#6}{}}{}{, #6}} & {\itshape #2}\\%
\end{tabular*}%
\ifx&%
\else{\\%
\begin{minipage}{\linewidth}% <=====================================
\small#7%
\end{minipage}}\fi%
\par\addvspace{#1}}
% https://tex.stackexchange.com/questions/351295/bullet-items-vertically-misaligned-if-moderncvstylecasual-is-used
\let\oldlbi\labelitemi
\renewcommand\labelitemi{\raisebox{.15em}{\oldlbi}}
\newcommand{\mycventrya}[4]{%
\renewcommand\labelitemi{\raisebox{0.8em}{\oldlbi}}
\begin{itemize}
\item{%
\mycventry{#1}{}{#2}{#3}{}{}
\vspace{-1.5\baselineskip}
\begin{changemargin}{0cm}{2.5cm}
#4
\end{changemargin}
}
\end{itemize}
\renewcommand\labelitemi{\raisebox{.15em}{\oldlbi}}
}
\name{John C.}{Lilly}
\address{On board research vessel \textit{Lusitania}}{}{}
\begin{document}
\makecvtitle
\section{Academic Qualifications}
\vspace{5pt}
\begin{itemize}
\item{
\cventry{1963--1967}{}{PhD Dolphinology}{University of Atlantis}{}{}
\vspace{-1.5\baselineskip}
\begin{changemargin}{0cm}{2.5cm}
Over the course of my research I became fluent in dolphin communication, developing a transcription method which I subsequently used in the writing of my thesis.
\end{changemargin}
}
\end{itemize}
\vspace{5pt}
\section{Publications}
\vspace{5pt}
\begin{itemize}
\item{
\textbf{Cetacean brains and higher-dimensional thought.}
My PhD thesis primarily outlines the contributions to the fields of multidimensional topology and geometry that can be gleaned from studying the latest discourses in dolphin mathematics.
}
\end{itemize}
\renewcommand\labelitemi{\raisebox{0.8em}{\oldlbi}} % <=================
\begin{itemize}
\item{
\cventry{1963--1967}{}{PhD Dolphinology}{University of Atlantis}{}{Over the course of my research I became fluent in dolphin communication, developing a transcription method which I subsequently used in the writing of my thesis.}
}
\end{itemize}
\renewcommand\labelitemi{\raisebox{.15em}{\oldlbi}} % <=================
\mycventrya{1963--1967}{PhD Dolphinology}{University of Atlantis}{Over the course of my research I became fluent in dolphin communication, developing a transcription method which I subsequently used in the writing of my thesis.}
\begin{itemize}
\item{
\textbf{Cetacean brains and higher-dimensional thought.}
My PhD thesis primarily outlines the contributions to the fields of multidimensional topology and geometry that can be gleaned from studying the latest discourses in dolphin mathematics.
}
\end{itemize}
\end{document}
you get the wished result:
