The typesetting of an image is handled exactly same way as that of a letter (say X)
This is line, first line.
X
Third Line.
Is a single paragraph equivalent to a source of
This is line, first line. X Third Line.
TeX may insert line breaks around the X or not according to its normal paragraph line breaking rules.
This is line, first line.
\begin{center}
X
\end{center}
Third Line.
Breaks the paragraph with a centered displayed X. Changing center to flushleft would be the same but flush left.
This is line, first line.
\begin{center}
X
\end{center}
Third Line.
is as above but now Third Line starts a new paragraph so may get different vertical spacing and/or indentation depending on the document settings.
If theimage is much larger than an X the usual form is to wrap it in
\begin{figure}
X
\caption{the letter x}
\end{figure}
Then LaTeX will "float" the figure to a suitable point to avoid bad page breaks.
Note that except in special environments like verbatim TeX always treats a single newline exactly like a space. So in your TeX source you can use newlines to improve the look.
An equation is typically entered
Some words
\begin{equation}
a + b
\end{equation}
more words
But TeX would see the same input from
Some words \begin{equation} a + b \end{equation} more words
or from
Some
words
\begin{equation}
a
+
b
\end{equation}
more
words
It is two consecutive newlines (a blank line) that causes different behaviour.
Some words.
More words.
is two paragraphs but while this typically forces a line break and possibly some additional vertical space and possibly indent More depending on the document class, it should be viewed as a logical markup of a paragraph not a way to force a vertical space in the output. It might be typeset as
¶Some words. ¶More words.
all on one line in some special document class layouts.