:salam2: brother
As far as I know, you can pray tahajjud at anytime during the night after 3eshaa of course.
But tahajjud is exclusively praying, and there are two opinions concerning it. The first is that it means praying at any time of the night, which is the view of the majority of fuqaha’.
The second is that it is prayer after sleeping. See: al-Mawsoo‘ah al-Fiqhiyyah, 2/232.
http://islamqa.com/en/ref/143240/tahajjud
But as you probably know, Allah descends to the lowest heaven during the last third of the night, as mentioned in the hadith qudsi below. So of course that's the best third of the night to make tahajjud.
It was narrated by al-Bukhaari in his Saheeh (1145) and by Muslim (1261) from Abu Hurayrah (may Allaah be pleased with him), that the Prophet (peace and blessings of Allaah be upon him) said:
“The Lord descends every night to the lowest heaven when one-third of the night remains and says: ‘Who will call upon Me, that I may answer Him? Who will ask of Me, that I may give him? Who will seek My forgiveness, that I may forgive him?’”
I'm not sure with regard to your question about how to say the ad3iyaa, but I know that you get hassanaat for speaking a good word and praising Allah, so why not say it out loud.
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These two videos below helped me a lot, when I was a bit confused about how and when to perform tahajjud. I hope it's beneficial to you as well.
[yt]ZiZLOfyCE4k[/yt]
[yt]RIyK3XmZeC4[/yt]
