a) $w$ will be CFL. Now how it will be CFL? Say grammar starts with 1.
So, first push 1,
Now, when 0 comes pop 1 with 0
if some extra 0 come push them in stack
then again when 1 comes pop 0 with 1
and like that we need to proceed, until it is empty stack.
[We can think grammar like this
$S\rightarrow 0S1|0S11|1S0|11S0|10S1|1S01|1|01|011|10|110|101$, but it is not a fully correct grammar]
It cannot be represented with a grammar
b)Complement of L will be $\left \{w|w\epsilon \left(0+1 \right ) ^{*}\right \}$ where $N_{0}\left ( w \right )>N_{1}\left ( w \right )$ or $N_{1}\left ( w \right )>2N_{0}\left ( w \right )$
c)yes it will be CFG.
Say grammar is $L=0^{n}1^{n}$
then $\frac{1}{2}L$ will be subset of $0^{n}1^{n}$. Now subset of CFG will be CFG or regular.
For example $L=0^{n}1^{n}$
and $1/2L=0^{n}$ which will be $0^{*}$ or $0^{*}1^{*}$
or it could be $0^{n/2}1^{n/2}$ , which is CFL
So, will be Regular or CFL
https://gateoverflow.in/93981/half-l