score:3
Buying a new ticket/supplement is for one necessary, if you want to use a train with compulsory seat reservations.
Example with a ticket A-B-C: If you miss the last ICE from B-C because the train A-B was delayed, and the only remaining possible train would be a Nightjet, you can use it, but you would have to pay the supplement first, and later reclaim it, since Nightjets have mandatory reservation.
Another usecase, as i mentioned in the comment on the other answer, if you have a ticket only for regional trains, and you want to use an ICE, then you also have to buy the ICE ticket and reclaim the costs.
Note: this applies only if the whole ticket itself is only valid for regional trains. If you have a ticket A-B-C where A-B is on an ICE, B-C would be a regional train, and you want to use an IC or ICE for B-C, because the first ICE was delayed, you don't have to buy an additional ticket for B-C, since your ticket is already an ICE ticket.
So regarding question 2, if you have a long distance leg on your ticket, you don't have to buy any new tickets, if you want to take IC/ICEs on originally booked regional train parts, if previous trains were delayed.
Upvote:5
The german version of the DB website gives some more details:
Wenn Sie eine Fahrkarte des Nahverkehrs (RE, RB, IRE und S-Bahn) besitzen, aber einen höherwertigen Zug (z.B. Züge des Fernverkehrs, also IC/EC, ICE) nutzen möchten, dann müssen Sie zunächst die zusätzlich erforderliche Fahrkarte bezahlen. Anschließend können Sie die Kosten [...] zurückfordern
Rough translation: If you have a ticket for a regional train (RE, RB, IRE, S-Bahn), but want to use a higher-class train (e.g. long distance trains, IC/EC, ICE), you have to buy the necessary ticket / upgrade at first. Later you can reclaim the cost.
So for the kind of journey you mentioned at the beginning of your question (journey with two legs), it doesn't depend on the kind of ticket you have, but on the kind of train you were supposed to take from B to C:
Unfortunately, I have no idea about the second case of your number 2 (journey with more than two legs).