Proving that $A=\{w_{2i}|w_{2i}\not\in L(M_i)\}$ is not Turing-recognizable
Lets reformat your language $A_D$ to say $A_D=\{ \langle M \rangle| \; \text{M is a Turing machine ( TM ) and does not accept }\langle M \rangle\}$. Here $\langle M \rangle$ means encoding of Turing machine $M$, and I assume the inputs are always valid encodings TM's ( as you can always check if a string is a valid encoding or not ). Now say there was a decider for $A_D$, let it be $D$. Now if you give input $\langle D \rangle $ to $D$ it has to halt ( we assumed it is a decider ) and either reject or accept. If it accepts it, it means $\langle D \rangle  \in A_D$ and thus $D$ does not accept $\langle D \rangle $, which is a contradiction. If $D$ reject $\langle D \rangle $ it means $D$ accepts $\langle D \rangle $ acccording to definition of $A_D$ which again is a contradiction. Either ways it is a contradiction, thus our assumption that $D$ exists is not correct and $A_D$ is undecidable ( this proof is related to Cantor's diagonal method ) .

This logic cannot be applied to prove $A_D$ is not  Turing recognizable. I leave the reason for you to figure out ( hint: no assumption about halting on all inputs is made ).

I will prove non recognizability  of a general language ( which implies undecidability  ) instead of $A_D$ and the one you specify in your question.
$L=\{  w | \; f(w) \;\text{ is a valid encoding of some Turing machine where } f \; \text{is some computable function and the Turing machine given by } \; f(w) \; \text{does not accept} \; w\}$ 
( $f$ is such that  for every TM encoding $\langle M \rangle \exists w $ s.t. $ f(w)=\langle M \rangle$ )
Now suppose $L$ was Turing recognizable, let the TM  be $R$. We construct another Turing machine $H$ which given an input $w$ first finds $f(w)$ ( again assuming it is a valid encoding of a TM, if not $H$ can directly reject ). Then $H$ gives $w$ to $R$ and simulates $w$ on $f(w)$ in parallel. If $f(w)$ accepts $w$ $H$ rejects, if $R$ accepts $w$, H accepts. This way you are able to decide language $L$ which is undecidable. You can prove $L$ is undecidable yourself ( see when $f$ is the identity function, $L$ reduces to $A_D$ which is undecidable ).  
I leave it for you to figure out how non recognizability of $L$ can be used for proving that of your languages.