Why would the Byzantines and the Russians be allied? Russia will still be expansionist. The Caucauses and the Dardanelles will be of interest to them just as the Baltic States, the Ukraine, and Poland were.
Common religion, common cultural influences, historical ties, already a history of working together for centuries, Russia's immense love of Byzantium, common dislike of steppe nomads (Not like the Ottomans who funded the Crimeans to harrass Russia), etc.
Not to mention geopolitically speaking, they both benefit. A friendly Byzantium gives Russia easy access to the straights and supplies from India, East Africa and Southeast Asia without much hassle, and probably at a lower cost than buying from the West. At the same time, a friendly Russia protects Byzantium from the various steppe nomads that move through Eurasia and allow it to better focus on other fronts that.
As to Austria, I have no idea. But a central European power would affect the Balkans. Whether tha power comes from Vienna, Prague, Budapest, Krakow, Venice doesnt really matter. The fact is the Byzantine Empire would have most likely faced a potent threat from their northwest.
And France faced Habsburg Encirclement. And the Dutch being in the middle of England, Spain and France. And Spain a coalition of powers determined to stop her at all costs.
In any case, with a friendly Russia the Byzantine Empire only has to focus on guarding her frontiers less. More forces can be put towards defending the Balkans.
This the number one challenge for them. Adaptation is actually incredibly hard. By 1700, the Byzantine Empire would have had 2000 years of tradition focusing their attention on the Levant, Balkans, and Mideast. Given the opportunities and threats that would have remained ITTL, looking beyond the region would have been highly unlikely.
What makes you say that? Because they never did it IOTL? States don't just stop having goals once they complete them. They find new ones.
And again, there's a tradition of crisis means reform. Adaption comes to the Byzantines somewhat more naturally than other states due to their historic experiences.
Colonial growth was a natural state for Britain, France, Netherlands, Spain, and Portugal because the lacked direct trade routes to the east. A Byzantine Empire would not.
The natural outgrowth past the Mideast would be India and the Far East; the sources of the Silk Road. Cutting out the middle man of Persia would be immensely profitable and beneficial to the state. Hence the establishment of trading outposts if need be.
I have no idea whether they would keep up with Western Europe or not. I would just note the further East one goes from the Rhine, the further these countries fell behind.
Determinism isn't useful.
The Turks had a remarkably diverse empire. Hell when they took Constantinople they had Venetians, Genoans, Austrians, Hungarians and a whole host of other non-Turks in their employment. That didnt help the Turks in the long run.
The Turks didn't mix nearly as much as the Byzantines did with their conquered subjects, nor did they move them around as much, nor were their government nearly as centralized for most of its history. Furthermore, again, the Ottoman state were much larger than the Byzantines for most of their history, making any changes harder to implement.
Looking at the Ottomans to find an analogy doesn't make much sense when you realize the significant differences in the way the governments worked, their cultures, the sizes and of course, their policies.
Perhaps the Byzantines would have been a colonial leader. But I dont think that TL is nearly as easy and likely as you do.
No effort at colonization is ever easy. Its always a significant investment filled with struggles.