بسم الله الرحمن الرحيم
(1) You gave one talaq on 28/05/2013 and after two days i.e. 30/05/2013 a child was born by operation. Did you take your wife back from the time of talaq to the child birth or not? If you took her back then the second talaq which you pronounced on 29/06/2013 took place and thus they became two talaqs in total. If you did not take her back i.e. you did not perform any act like intercourse or kiss etc and also you did not say verbally that you are taking her back, if it is so then the second talaq which you gave on 29/06/2013 was invalid, as the iddah of first talaq was over due to child birth. And when you gave second talaq then she was no longer in your nikah.
(2) There is no need and also it should not be given. In case the second talaq occurred and the iddah was over then the divorced woman shall have the right to marry where she wishes.
(3) You should inform your wife and her parent in clear cut words.
(4) The expenses of iddah are wajib on you; if you do not give it then too talaq shall not turn invalid.
(5) Would you have given talaq through any Darul Qaza it would have provided you the same. Consult any Darul Qaza near you.
(6) No time and day is fixed for it. You may do it when you wish.
(7) Now it is not necessary to take permission from anyone as per the Shariah. However you should seek the advice of your parents and relatives.
Allah knows Best!
Darul Uloom Deoband