Is it in any way possible for enough male members of the House of Hohenzollern to die or be born girls for a member Hohenzollern-Sigmaringen to become King of Prussia and eventually Kaiser of the German Empire?
They are Catholic, so a Sigmaringen heir has to convert first to Protestantism.
Would they even have a claim to Prussia (and Brandenburg)? IIRC the division between the Sigmaringen and the Brandenburg branches goes back to the 13th century, and the Sigmaringen was the senior line of the family. They didn't have any prior claim to Brandenburg (and after to Prussia) as those territories were given to the junior line almost 200 years after their division.
That's how the Brandenburg got Prussia in the first place.
In 1871, there were in order of succession:
1. Wilhelm I. (regnant)
2. Friedrich (III.), Son of 1
3. Wilhelm (II.), Son of 2
4. Heinrich, Son of 2
5. Waldemar, Son of 2
6. Carl, Brother of 1
7. Friedrich Karl, Son of 6
8. Friedrich Leopold, Son of 7
9. Albrecht (until 1872), Brother of 1
10. Albrecht, Son of 9
11. Alexander, Great-Grandson of King Wilhelms Grandfather FW III
12. Georg, Great-Grandson of King Wilhelms Grandfather FW III
(By 1880, 10 will have three sons.)
That is quite a massacre.
They might to jump to the Swabian branch, whose head in 1871 is
Sw1. Prince Karl Anton (dies 1885)
Sw2. Leopold, Son of Sw1
Sw3. Wilhelm, Son of Sw2
Sw4. Ferdinand, Son of Sw2
Sw5. Karl Anton, Son of Sw2
Sw6. Karl, Son of Sw1
Sw7. Friedrich, Son of Sw1
But generally, there is the problem that without a crazy "King Ralph" extinction event, when a monarch proves presumably childless, with the Hiohenzollern there are always younger satill living brothers who can be made to marry some princess. Killing the dynasty off is hard. Especially has there is also the Franconian branch in Ansbach and Bayreuth ...
Would they even have a claim to Prussia (and Brandenburg)? IIRC the division between the Sigmaringen and the Brandenburg branches goes back to the 13th century, and the Sigmaringen was the senior line of the family. They didn't have any prior claim to Brandenburg (and after to Prussia) as those territories were given to the junior line almost 200 years after their division.
Would that necessarily matter?
It was a pretty absolute kind of monarchy, and if the last King in the old line named the elder branch as his heirs, that would probably be made to stick - esp if he'd got his army officers to take and oath to uphold the settlement.
And if the male line looked like failing in this way, the nearest female relative could be married off to a Sigmaringen.