Do we have to say 'Allahu Akbar' loudly in every rakat of every salat (prayer)? Or is it necessary in some specific salat or rakat?
It is not considered as a Wajib (obligatory) act to say it loudly in every Rekat. Besides, Takbiratol-Ehram is considered as a Mustahab or recommended practice for the Imam to say it loudly that the persons who prayer, be able to hear it. But it doesn’t seem to be counted as a Wajib or obligatory act.
In regard to saying it in other parts of the prayer, we should notice that those are not Wajib practices to say them loudly such as those Takbirs (Allah-o-Akbars) which are related to Rokus and Sojud.
According to Grand Ayatollah Sistani as a Marja’-al-Taghlid of Shia:
You ought to say it (Takbirat-al-Ehram) somehow that at least you (yourself) be able to hear your voice… (www.sistani.org)
That depends on what you mean with loud?
An Imam should do this as his followers should follow him. So this is a way to communicate therefore he must communicate the transfer with an appropriate voice (so a more loudly voice would be mandub for him only in all the takbirat)!
(Takbir/takbira(t) =(to say) Allahu Akbar)
As a fad (praying alone) or a Ma'mum in general you won't be asked to say it loudly, but some do it. As you are free to say it in the manner and ton you feel comfortable with. But most scholars would say a lower voice would be preferable and a louder makroh! One could take as a reference this Hadith from Sunan abu Dawod but there's a better or more significant narration in Musnad al Imam Ahamd:
أَمَا إِنَّ أَحَدَكُمْ إِذَا قَامَ فِي الصَّلَاةِ فَإِنَّهُ يُنَاجِي رَبَّهُ ، فَلْيَعْلَمْ أَحَدُكُمْ مَا يُنَاجِي رَبَّهُ ، وَلَا يَجْهَرْ بَعْضُكُمْ عَلَى بَعْضٍ بِالْقِرَاءَةِ فِي الصَّلَاةِ
When one of you stands to pray, he is conversing with his Lord, so let each of you think of what he is saying to his Lord, and no one among you should raise his voice over anyone else when reciting in the prayer.
As a woman the loudest voice you could use is that if somebody who's next to you could hear a low sound but not follow your words.
In a congregation sometimes one ma'mum is asked to repeat the takbir after the Imam in a loud voice as the mosque might be too large to let everybody hear the Imam!
Among the Sunni sects the Maliki madhab is the only one where it is preferable (mandub) not only for the Imam to do the takbirat al Ihram (the takbir to enter/start the prayer) in a loud voice as it is the case for Shi'a.
And Allah knows best!