Is KJ-600 will operated in Liaonoing (CV-16) and Shandong (CV-17)?
We don't know.
Theoretically-speaking, the angled landing strips on both Liaoning and Shandong are wide enough to enable KJ-600 doing BAR operations. However, it's not just whether the KJ-600 is able to conduct STO operations onboard the STOBAR twins, but also whether both ships have sufficient deck spaces and necessary facilities to accommodate them are entirely different stories.
And so far, we haven't heard of any positive indications of KJ-600 making it onboard the STOBAR twins, so there's that.
Any rumours recently about when work will start on 052D/055 successors? Saw some comments on weibo especially for the 9000t that construction may begin in the next few months.
R&D works on the next-generation successors to the major surface combatants (DDGs and FFGs) of the PLAN have been going on for quite some while by now.
As for the news/rumors of construction works being started - There has been claims of "movements" in Hudong-Zhonghua and Huangpu-Wenchong, where both shipyards are known for building FFGs for the PLAN. Whether those WIP warships are new 054B FFG units or belong to a brand-new FFG class remains to be seen.
Either way, you can refer to the Next-Generation DDG/FFG thread for more information.