What the relationship with western countries specially trading rivals ?
In the late 13th and early 14th centuries, the Silk Roads were by far the richest trade routes in the world. There were three major routes, the Chang'an/Samarkand-Antioch line (Golden Road), the Chengdu-Alexandrian line (Incense Road), and the Chang'an-Tmutarakan/Trapezous line (Old Road). While the Golden Road was by far the richest of the routes, the Chagatai, Yuan and Il-Khanates were involved in a three-way struggle, resulting in most trade going through Alexandria and Trapezous. The current Mameluke sultan, al-Nasir Mohammed, was the Mad Caliph on steroids, and refused the Italian traders access to the ports of Alexandria.
By 1305, Trapezous and Tmutarakan processed roughly about 70-80% of all trade between the Orient and Europe, which obviously gave the Romans a crap ton of influence in international trade. Of Frei and Alexios took full advantage of that, barring Venetian and Genoese traders from their ports, encouraging the smaller republics of Pisa, Ankona and Ragusa against them. The two rival banded together against the Romans several times, but internal infighting always destroyed them before they became large enough to pose a threat to the Romans. So both the Venetians and Genoese hate the Romans, but this was balanced out by strong influence with the rising powers of Ankona and Pisa.
Moving outside of Italy, the Romans have had fairly positive interactions with the English, as the fabric manufacturers of Namen have good relations with their producers across the channel. This obviously only worsened their relations with Phillip IV, bringing the two groups to the brink of conflict in 1301, when French troops briefly occupied Namen. This spooked the electors into throwing together a coalition of 30,000 men and marching to meet Phillip. King Albert threatened to bring in the English and Bohemians, which would bring tensions to a boiling point and start a pan-European War, which no one wanted.
Speaking of tensions, in 1305 Antipope Nicholas V was elected in Rome in opposition to Clement V in Avingnon. Nicholas V crowned King Albert of Germany as Holy Roman Emperor in 1305, ending the fifty-year Great Interregnum. This really pissed off Phillip IV, but he couldn't do anything about it as deposing Nicholas V would lead to war with the HRE, which would open up a flank to the English, which would in turn open up an opportunity for the Socts to invade, which would then do the same for the Norweigans.
This July Crisis-esque scenario brings us to Europe's most eligible bachelor, Wenceslaus the Thrice-Crowned. OTL Wenceslaus was a middling ruler, but in this TL he's a Peter the Great-esque great ruler, which just makes everyone else more nervous as Wnceslaus is the King of Hungary-Croatia-Bohemia-Poland. (Insert the I fear no man meme) As of 1305, Andronikos Raoul, Alexios' nephew, is betrothed to Wenceslaus' illegitimate daughter Elizabeth. However, there is still serious competition to maneuver a dynast into a marriage with the Thrice-Crowned.
The Hungary-Rome alliance has effectively convinced Stefan Milutin to just say no and stop making trouble on either border. It has also lead Yuri, the theoretical King of Ruthenia and Alejsi of Bulgaria to form an alliance to keep the allies off their back.
The Ilkhanate is enough of a hot mess that I'm going to go over it separately tomorrow.