How to get numeric value from array

You can use more flexible methods; arrayjobx is unnecessarily complex and expl3 methods are much more powerful.

\documentclass{letter}
\usepackage{xparse}

\ExplSyntaxOn
\NewDocumentCommand{\setarray}{O{;}mm}
 {
  \seq_clear_new:c { g_alexb_array_#2_seq }
  \seq_gset_split:cnn { g_alexb_array_#2_seq } { #1 } { #3 }
 }
\NewDocumentCommand{\getfromarray}{mm}
 {
  \seq_item:cn { g_alexb_array_#1_seq } { #2 }
 }
\cs_generate_variant:Nn \seq_gset_split:Nnn { c }
\cs_set_eq:NN \inteval \int_eval:n
\ExplSyntaxOff

\newcounter{ExpOneTotal}

\begin{document}

\setarray{First}{10;8;3}
\setarray{Second}{1;7;8} 
\setarray{Third}{1;5;16}

\begin{tabular}{|c|c|c|c|c|}
\hline
Experiment & a & b & c & TOTAL \\ \hline 
Exp 1 & \getfromarray{First}{1} & \getfromarray{First}{2} & \getfromarray{First}{3} &  \\ 
Exp 2 & \getfromarray{Second}{1} & \getfromarray{Second}{2} & \getfromarray{Second}{3}  & \\ 
Exp 3 & \getfromarray{Third}{1} & \getfromarray{Third}{2} & \getfromarray{Third}{3} & \\ 
TOTAL & & & & \\ 
\hline 
\end{tabular}

\getfromarray{First}{1}

\the\numexpr \getfromarray{First}{1} \relax 

\addtocounter{ExpOneTotal}{\numexpr \getfromarray{First}{1} \relax}

My counter is \theExpOneTotal{}; should be \getfromarray{First}{1}.

\setcounter{ExpOneTotal}{%
  \inteval{
   \getfromarray{First}{1}+
   \getfromarray{Second}{1}+
   \getfromarray{Third}{1}
  }
}

My counter is \theExpOneTotal{}; should be
  \inteval{
   \getfromarray{First}{1}+
   \getfromarray{Second}{1}+
   \getfromarray{Third}{1}
  }

\end{document}

As you see, with \inteval you don't even need to set up a counter, because the computation can be done on the fly.

In the \setarray command the delimiter is customizable, you can do

\setarray[&]{First}{10&8&3}

if you prefer (or even change the token for the default value of the O argument).

You can even use floating point values, using \fpeval instead of \inteval (requires loading the xfp package).

\First is not expandable and can therefore not be used in \numexpr. The package arrayjobx provides the following alternative:

\check<arrayname> extracts the desired value and stores it in \cachedata that can now be used in \numexpr and other places:

\documentclass{letter}
\usepackage{arrayjobx}

\begin{document}

\newarray\First
\readarray{First}{10&8&3}
\newcounter{ExpOneTotal}

\First(1)

\checkFirst(1)\cachedata~=~\the\numexpr\cachedata\relax

\addtocounter{ExpOneTotal}{\cachedata}

My counter is: \theExpOneTotal~=~\First(1)

\end{document}

With calculations of row and column sums:

\documentclass{letter}
\usepackage{arrayjobx}
\usepackage{booktabs}

\begin{document}

\newarray\First
\newarray\Second
\newarray\Third 
\readarray{First}{10&8&3}
\readarray{Second}{1&7&8} 
\readarray{Third}{1&5&16}
\newcounter{ExpTotali}
\newcounter{ExpTotalii}
\newcounter{ExpTotaliii}

\def\AddTotal#1(#2){%
  \csname check#1\endcsname({#2})%
  \cachedata
  \addtocounter{ExpTotal\romannumeral#2}\cachedata
}

\newcounter{ExpTotalSum}
\newcounter{ExpSum}
\makeatletter
\def\ExpSum#1(#2){%
  \setcounter{ExpSum}{0}%
  \begingroup
    \@for\I:=#2\do{%
      \csname check#1\endcsname(\I)%
      \addtocounter{ExpSum}{\cachedata}%
    }%
  \endgroup
  \theExpSum
  \addtocounter{ExpTotalSum}{\value{ExpSum}}%
}
\makeatother

\begin{tabular}{ccccc}
\toprule
Experiment & a & b & c & TOTAL \\
\midrule
Exp 1 & \AddTotal{First}(1) & \AddTotal{First}(2) & \AddTotal{First}(3) &
  \ExpSum{First}(1,2,3) \\ 
Exp 2 & \AddTotal{Second}(1) & \AddTotal{Second}(2) & \AddTotal{Second}(3) &
  \ExpSum{Second}(1,2,3) \\ 
Exp 3 & \AddTotal{Third}(1) & \AddTotal{Third}(2) & \AddTotal{Third}(3) &
  \ExpSum{Third}(1,2,3) \\ 
\midrule
TOTAL & \theExpTotali & \theExpTotalii & \theExpTotaliii &
  \theExpTotalSum \\ 
\bottomrule
\end{tabular}
\end{document}

With alignment of the numbers via package siunitx:

\documentclass{letter}
\usepackage{arrayjobx}
\usepackage{booktabs}
\usepackage{siunitx}

\begin{document}

\newarray\First
\newarray\Second
\newarray\Third 
\readarray{First}{10&8&3}
\readarray{Second}{1&7&8} 
\readarray{Third}{1&5&16}
\newcounter{ExpTotali}
\newcounter{ExpTotalii}
\newcounter{ExpTotaliii}

\def\AddTotal#1(#2){%
  \csname check#1\endcsname({#2})%
  \addtocounter{ExpTotal\romannumeral#2}\cachedata
  \global\let\CD\cachedata
}

\newcounter{ExpTotalSum}
\newcounter{ExpSum}
\makeatletter
\def\ExpSum#1(#2){%
  \setcounter{ExpSum}{0}%
  \begingroup
    \@for\I:=#2\do{%
      \csname check#1\endcsname(\I)%
      \addtocounter{ExpSum}{\cachedata}%
    }%
  \endgroup
  \addtocounter{ExpTotalSum}{\value{ExpSum}}%
}
\makeatother

\begin{tabular}{c*{4}{S[table-format=2]}}
\toprule
Experiment & a & b & c & TOTAL \\
\midrule
Exp 1
  & {\AddTotal{First}(1)}\CD
  & {\AddTotal{First}(2)}\CD
  & {\AddTotal{First}(3)}\CD
  & {\ExpSum{First}(1,2,3)}\theExpSum \\ 
Exp 2
  & {\AddTotal{Second}(1)}\CD
  & {\AddTotal{Second}(2)}\CD
  & {\AddTotal{Second}(3)}\CD
  & {\ExpSum{Second}(1,2,3)}\theExpSum \\
Exp 3
  & {\AddTotal{Third}(1)}\CD
  & {\AddTotal{Third}(2)}\CD
  & {\AddTotal{Third}(3)}\CD
  & {\ExpSum{Third}(1,2,3)}\theExpSum \\ 
\midrule
TOTAL & \theExpTotali & \theExpTotalii & \theExpTotaliii &
  \theExpTotalSum \\ 
\bottomrule
\end{tabular}
\end{document}