Sentences are crossing the margin. Is there any solution?

Adding \usepackage[latin]{babel} will also help in your case.

\documentclass[12pt,a5paper,twoside]{book}
\usepackage{ebgaramond} % main font
\usepackage[utf8]{inputenc}
\usepackage[latin]{babel}
\usepackage{blindtext}
\usepackage{xcolor}
%%%%%%%%%%% MARGIN %%%%%%%%%
\usepackage[left=15mm,right=15mm,top=20mm,bottom=20mm,headheight=0mm,]{geometry}
\renewcommand{\baselinestretch}{1.2}
%%%%%%%%% LETTRINE %%%%%%%%%%
\usepackage[T1]{fontenc}
\usepackage{lettrine}
\input Zallman.fd %http://ctan.forsale.plus/fonts/initials/Zallman.fd
\newcommand*\initfamily{\usefont{U}{Zallman}{xl}{n}}
%letterine setup
\setcounter{DefaultLines}{4}
\renewcommand*{\DefaultLoversize}{0.1}
\renewcommand*{\DefaultLraise}{-0.07}
\setlength{\DefaultFindent}{0.8mm}
\setlength{\DefaultNindent}{0em}
\renewcommand{\LettrineFontHook}{\initfamily\color{green!0!black}}
%%%%%%%% BEGIN DOCUMENT %%%%%%
\begin{document}
\chapter{Main Heading}
\lettrine{A}{nd} \blindtext
\blindtext
\blindtext
\end{document}

The solution here is usually to enforce the hyphenation of the faulty words.

Since you are using a blind text, some words are not properly hyphenated.

In your case, adding \hyphenation{Do-nec} to the preamble solves the problem (I don't know if that's the correct hyphenation, though).

\documentclass[12pt,a5paper,twoside]{book}
\usepackage{ebgaramond} % main font
\usepackage[utf8]{inputenc}
\usepackage{blindtext}
\usepackage{xcolor}
%%%%%%%%%%% MARGIN %%%%%%%%%
\usepackage[left=15mm,right=15mm,top=20mm,bottom=20mm,headheight=0mm,]{geometry}
\renewcommand{\baselinestretch}{1.2}
%%%%%%%%% LETTRINE %%%%%%%%%%
\usepackage[T1]{fontenc}
\usepackage{lettrine}

\hyphenation{Do-nec}% <--- HERE

\input Zallman.fd %http://ctan.forsale.plus/fonts/initials/Zallman.fd
\newcommand*\initfamily{\usefont{U}{Zallman}{xl}{n}}
%letterine setup
\setcounter{DefaultLines}{4}
\renewcommand*{\DefaultLoversize}{0.1}
\renewcommand*{\DefaultLraise}{-0.07}
\setlength{\DefaultFindent}{0.8mm}
\setlength{\DefaultNindent}{0em}
\renewcommand{\LettrineFontHook}{\initfamily\color{green!0!black}}
%%%%%%%% BEGIN DOCUMENT %%%%%%
\begin{document}
\chapter{Main Heading}
\lettrine{A}{nd} \blindtext
\blindtext
\blindtext
\end{document}