As for Russian Empire (which was Muscovy by then) not only they tried to declare themselves successor of Byzantium, they also invented a claim that rulers Ivan IV and Vasiliy III were descendants of a Roman Emperor Augustus.

Both claims are, let's say, very disputable.

I don't know how much the people of the Ottoman Empire associated themselves with the Roman Empire, but the emperors did consider themselves successors starting from the annexation of Constantinople. This can be seen in the title Kayser-i Rum(Ceasar of Rome) that the Ottoman emperors beginning with Mehmet II use in their letters and edicts. I am currently on mobile at school but I can give you sources later on. Until then, you can research the title Kayser-i Rum and I think you will be able to find sufficient information online.

The religious factor is paramount here: after the fall of Rome, Constantinople was considered the successor of the 'true' Christian legacy. This was followed by the schism between Eastern and Western churches hence the un-reconciled rift between Orthodoxy and Catholicism. The missionary presence in the Balkans and Russia the 10th century resulted in the conversion of Kievan Rus (then the political center of Russia). After the conquest of Constantinople in 1453 by the Ottomans, Russia assumed the heritage of the 'true' christian legacy as it was Orthodox and un-reconcilable with Catholicism and Constantinople the previous heir had collapsed. Therefore Russia styled itself as a 'Third Rome' and the 'true' successor of Christendom.

The connections as you can see are somewhat tenuous and embellished. In reality there was probably nothing similar between Rome on the eve of its demise and Russia when it assumed the title of 'Third Rome'. The only real connection between all three is the belief that they are the inheritors of the 'true' Christian Church which was founded in Rome.

As is known, the Holy Roman Empire also laid claim to the legacy of Rome, and after the fall of Constantinople, the Habsurgs also considered themselves successors of Rome for a time. Their claim was not as well established or in use as the claims of the Ottomans and Russians though.

The Ottoman Turks over through Byzantium after the siege of Constantinople in 1453 and thereby considered themselves the successors of the Rome after having defeated the last reminance of Rome. Russia had a Czar or as I prefer Tsar which is Russian for Caesar. Idk if that's how they became such big enemies against each other or not though.

Russians considered themselves not exactly as successors of Roman Empire, but as successors of Byzantine Empire. Moscow is called "Third Rome" for a reason. It is also believed that the name "Tsar" was derived from "Caesar".