Statement of Purpose Template

Replace the center environment with the \centering switch to avoid additional vertical space. For the second issue simply use small capitals for the title. See code below for the modifications.

\documentclass{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[margin=1in]{geometry}

\newcommand{\HRule}{\rule{\linewidth}{0.5mm}}
\newcommand{\Hrule}{\rule{\linewidth}{0.3mm}}

\makeatletter% since there's an at-sign (@) in the command name
\renewcommand{\@maketitle}{%
  \parindent=0pt% don't indent paragraphs in the title block
  \centering
  {\Large \bfseries\textsc{\@title}}
  \HRule\par%
  \textit{\@author \hfill \@date}
  \par
}
\makeatother% resets the meaning of the at-sign (@)

\title{Statement of Purpose}
\author{name}
\date{Ph.D. Applicant}

\begin{document}
  \maketitle% prints the title block

  Emergency medicine has always been a passion of mine\ldots
\end{document}

For the extra line, you can tweak it manually by using negative vertical spaces :

\vspace{-0.2in}

However, be careful as this can mess up your document if you give too big numbers.