You would have to have a PoD in the 17th century or so, plus a complicated set of marriages and inheritances. Quite likely this results in an unrecognisable Europe.
Stability/Centralising: These two are basically mutually exclusive for an empire of this size. The OTL AHE tried focussing everything into Vienna and Budapest, and the dozens of minorities just tore the whole thing apart. If Berlin is the power base, a similar outcome would happen. I can't imagine any other cities ruling that empire.
Colonies: Has no reason to take them. For one, its coastline is rubbish, being locked up in the Baltic and Mediterranean. As I said above, it would also struggle with internal problems so would not be in a good position to start taking more land.
Army, Navy: The German High Seas Fleet was a white elephant in real life, and as a continental power this country would not need much of a navy anyway. However, the army would almost certainly be large. How large would depend on the conscription laws (assuming conscription comes into being in this TL), and the quality would depend on if it adopts the Prussian methods. Although, without the Austro-Prussian War in 1866, the Prussians have less opportunity to prove that their methods are the superior ones.
Rich: Large tax base, also very large expenditure. Can't say much else with any degree of certainty.
Rivals: Excepting a modernised Russia, no country in Europe could hope to challenge it, much less be a rival. OTL Germany alone was able to hold off most of Europe for a long time in both World Wars, so adding another country of Germany's size, with German level organisation, would probably be more than a match for anyone. Likely including America.
WWI: This was butterflied long ago (as I said, 17th century). Maybe a quick war between Russia or America and Uber-Germany, but no 4-year conflict.
- BNC