Multilingual Title Page with Babel?

This is an interesting issue.

First of all, as has been pointed out in the comments by Cicada, you will want to change the language for your author, since it is in English. If the language is correctly switched, the encoding will also be switched allowing you to use th (or even þ - at least in modern versions of LaTeX) without problems.

The problem is that the naive approach of

author{%
  foreignlanguage{USenglish}{%
    Typeset by:
    Matþew Scarbrough}}

does not work as expected.

To understand why, we need to dig a little deeper in the macros that article uses to typeset the title.

You probably know that author is defined (in latex.ltx) as

DeclareRobustCommandauthor[1]{gdef@author{#1}}

so author{<author>} just saves <author> in the internal macro @author. So far so good. With your settings (the default notitlepage) maketitle does some preparation and then calls @maketitle to do the typesetting of the title. @maketitle is defined (in article.cls) as

def@maketitle{%
  newpage
  null
  vskip 2em%
  begin{center}%
  let footnote thanks
    {LARGE @title par}%
    vskip 1.5em%
    {large
      lineskip .5em%
      begin{tabular}[t]{c}%
        @author
      end{tabular}par}%
    vskip 1em%
    {large @date}%
  end{center}%
  par
  vskip 1.5em}

This means that the author (as given in @author) is typeset in a tabular environment.

In particular

author{%
  foreignlanguage{USenglish}{%
    Typeset by:
    Matþew Scarbrough}}

Will cause LaTeX to typeset

begin{tabular}[t]{c}%
  foreignlanguage{USenglish}{%
    Typeset by:
    Matþew Scarbrough}
end{tabular}

Or arranged more suggestively

begin{tabular}[t]{c}%
  foreignlanguage{USenglish}{Typeset by:
  Matþew Scarbrough}
end{tabular}

Indeed, the starts a new row in the tabular environment and not 'just' a new line as one might have expected. Since each cell is in a group of its own, this causes issues with missing closing and opening braces. You can verify that a command in the first 'row' does not affect the second row with

author{%
  color{red}This is red
  This isn't red
}

Furthermore, everything works absolutely fine if you drop the

author{%
  foreignlanguage{USenglish}{Typeset by: Matþew Scarbrough}%
}

A simple solution then is to issue foreignlanguage for each 'row' separately in author.

documentclass[letterpaper]{article}
usepackage[T1,T2A]{fontenc}
usepackage[USenglish,main=russian]{babel}
usepackage{xcolor}
usepackage{indentfirst}

addtocaptionsUSenglish{%
  renewcommand{contentsname}{Table of Contents}%
}

title{%
  Учимся говорить по-алтайски.
  Урок № 1. Алфавит.%
}

author{%
  foreignlanguage{USenglish}{Typeset by:}
  foreignlanguage{USenglish}{Matþew Scarbrough}%
}

begin{document}
pagenumbering{roman}
maketitle

tableofcontents
newpage

pagenumbering{arabic}
section{<Раздел>}
Съешь же ещё этих мягких французских булок да выпей чаю...
end{document}