As for Charles Anjou oh yeah of course his successors would want to claim Nicea and Trebizond but the thing is by this point if he controls latin empire+the rich french counties+rich provence etc as well as much of Italy and Southern Italy well the guy is going to be in charge of quite the strong nation which means he may be viewed as a big threat so I think he will by that point or his successors be more focused on managing the nobles across his scattered lands and protect them from ambitious french kings and HRE. So I dont think he will make a play for Anatolia his and his successors goals will be to control Italy and the balkans and Greece and maybe if possible also much of southern france and possibly Aragon since he had a not so good relationship with the Aragonese given the Aragonese were closely tied to his hated enemies the House of Staufen. He may make a play at Jerusalem but again he and his successors will have too much on their plate to handle and I expect eventually a conflict between Anjou and Capet/Valois in later generations since by that point Anjou would have become to powerful for the French King to allow it to do whatever it wants.
Komennoi doukai wont make a play for Nicea he be too busy fighting the serbs, bulgarians, latins, franks, pretenders, etc. Unfortunately in a nonasb situation he is pretty much screwed unless things go horribly wrong for all his neighbors. I think if he is able to stabilize the empire and by some miracle defeat all the myriad of western foes then he may focus on the east but by then I think Nicea would be too powerful. Vatatzes was a brilliant guy in a different sense from Michael VIII. Wheras Michael enjoyed support among the elite byzantine families, Vatatzes was more of a populist and his reforms were aimed at strenghtened the power of the beurocracy at the expense of the byzantine nobility. One of the reasons Michael took power otl was that the vatatzes pissed off to many nobles, but in this timeline if Vatatzes focuses on Anatolia to build a power base he would have much greater success and can install loyalists in the newly conquered territories and that coupled with popular support would make any serious noble revolt unlikely. If Nicea can take trebizond and foster a good relationship with the Ilkhanate perhaps as a counter to the mamluks then he can take advantage of the profitable silk road trade route. In such a case the komenoi doukas are done for, facing a power Nicean state on one side and western christendom as well as angry slavic neihbors from the other side. I think what will most likely happen is that Komenoi doukai may engage in a political marriage with a powerful Vatazes in the best case and reconcile east and west that way though in the worst case I see a partition of the Komnenoi doukai lands, perhaps exhausted from his wars with the western powers the Niceans take advantage and conquer constantinople. Things were that bad unfortunately for Theodore Komnenois Doukas.
On the other hand if Theodore is able to overcome his enemies maybe he conquers bulgaria and serbia, drives the franks from Greece and a host of lucky events happen that cause his powerful neighbors to weakean he could perhaps in a best case retake much of the balkans sans Croatia and Greece though maybe the pelopenese would be difficult. In any case such a strong state could contend with the Nicean state but even then the problem is the lack of a navy. The Italians including Venice will not want this rising state to have a strong fleet to challenge their dominence and that means even if Theodore wants to invade Nicea he wont have a fleet capable of transporting the invasion force neccessary to defeat the Niceans and vice versa. For him to make a go at Nicea at the very least the Italian martime powers need to be checked and the Byzantine navy strong enough to repel them a big call.
