Do the corrupted Grail fragments that Zokuen synthesised with his worms which were later placed in Sakura have a direct connection with Angra Mainyu inside the Grail? I know the fragments are the remnants of the Lesser Grail of the 4th HGW and are corrupted because Angra used the Lesser Grail to activate the Greater Grail and unleash his curses but can he control the fraagments after they were implanted inside the worms? And what about Illya's heart which is the Lesser Grail of the 5th war?
Do the corrupted Grail fragments that Zokuen synthesised with his worms which were later placed in Sakura have a direct connection with Angra Mainyu inside the Grail?
yes, in the Visual Novel close to the end
after Sakura realizes the love Rin has for her she starts trying to resist. when Shirou arrives after defeating Alter Saber he finds the "ribbons" that make up Dark Sakura's dress are now being used to wrap her up and restrain herr. since at this point Dark Sakura has already pruged all of Zouken's worm from her body the only force that could do this is Angra Mainyu, further enforced by the fact that they dispelled when Shirou stabbed Sakura with the traced Rule Breaker, severing the contract between her and Angra Mainyu
however she probably need to submit to Angra Mainyu as
Dark Sakura doesn't appear until after she kills Shinji (to tried to rape and blackmail her) and realizes she was The Shadow
and up to then she didn't even want to fight in the war
however this is only the case because Sakura is a Master. in Fate and Unlimited Blade Works, as she gives it up to Shinji and in both cases Rider is considered defeated without Sakura's identity as her True Master being revealed one can assume that she had fully given up being a Master in these 2 routes while in Heaven's Feel she hadn't and it's only in this route does the Shadow appear. also during the period between the 4th and 5th wars Angra Mainyu didn't do anything with Sakura.
And what about Illya's heart which is the Lesser Grail of the 5th war?
supposedly uncorrupted. given that the main source of the corruption is the Greater Grail and the Einzbern knew of it's state (as it was Ilya who revealed Angra Mainyu's existence to Rin and Shirou) it would not be hard for them to create an uncorrupted lesser grail, however this would not stop them connecting to the corrupted Greater Grail as Irisviel's lesser grail connected to it, so i'm not sure how the Einzberns planned to get around Angra Mainyu to get the 3rd Magic (or maybe they would just accept the destruction because all they care about is reclaiming it)
Angra used the Lesser Grail to activate the Greater Grail and unleash his curses
actually he didn't. if he could his wish to be born could have been done without anyone getting in the way, but while he is not manipulating Kotomine he did need him to wish for his birth, and by his own will Kotomine wants Angra Mainyu to be born.
just like in Fate and Unlimited Blade Works, the connection between the lesser grail and greater grail is the reason for the black mud to be spilling out and in Fate/Zero the fire caused by the spilled contents is hinted to be because Kotomine, before dying, wished for humanity to die aswell
In Kirei's last moments, the Holy Grail appears to him in his mind. Filled with hatred and envy for Kiritsugu, Kirei reaches out and touches the cursed artifact. He prays for an end to humanity before his death and his wish is granted in the form of the fire.
Source: Avenger (Fate/Stay Night) > Role > Fate/Zero (6th paragraph)