Refinement of nested cases / aligned equations

Here's a solution to your second question. AFAICT, the solution answers also your first question implicitly.

Some comments:

• I added a left( ... right) "wrapper" around the first fraction term, as otherwise it's not obvious whether the nu exponent term pertains to the entire fraction or to just the numerator.

• The solution also inserts typographic struts in the second "inner" array to assure that the rows' heights are the same as in the first "inner" array.

• Finally, addlinespace directives (provided by the booktabs package) are inserted to increase the vertical separation between rows.

That said, I don't believe that this layout is all that clear. For instance, shouldn't there be an indication somewhere that rows 2 and 3 pertain to the case $nu le 1?

documentclass{article} 
usepackage{nccmath}   % for 'fleqn' env.
usepackage{array}    % for newcolumntype macro
usepackage{booktabs} % for addlinespace macrdo
newcolumntype{L}{>{displaystyle}l}
% Define two (typographical) struts (to be used in math mode only):
newcommandstrutA{vphantom{%
   displaystyleleft( frac{k}{N_0} right)^{nu}}}
newcommandstrutB{vphantom{%
   displaystylefrac{k-N_0}{N-N_0}}}
   
begin{document}
begin{fleqn}
begin{equation}
t_k = 
    left{
    begin{array}{@{} L @{quad} L @{}}
        left(frac{k}{N}right)^{!nu} 
        & text{for $nu > 1$} 
        addlinespace
        left{ 
        begin{array}{@{} L @{}}
           t_0 left(frac{k}{N_0}right)^{!nu} 
           addlinespace
           t_0 + frac{k-N_0}{N-N_0} (t - t_0)
        end{array}
        right.kern-nulldelimiterspace
        &
        begin{array}{@{} L @{}}
           text{for $0 leq k leq N_0$}strutA  
           addlinespace
           text{for $N_0 < k leq N$, $t_0 < t$}strutB
        end{array}
    end{array}
    right.
end{equation}
end{fleqn}
end{document}