edit: it looks like Deutsche Bahn changed how they issue tickets from/to cities with Cityticket agreement. If a (Super) Sparpreis ticket starts/ends with a local train trip to/from a long-distance train in the same Cityticket area, this trip is now always included as a Cityticket.
Unfortunately not
For a long time Deutsche Bahn offered the so-called "tarifliche Gleichstellung", which grouped the stations of a city into a larger virtual station (e.g. Berlin Central Station, Berlin Südkreuz etc. were grouped into the BERLIN virtual station) - a ticket from/to such a grouped station was valid from/to all stations in the group. However, they stopped offering it for long-distance tickets in December 2023. Since then you're required to adhere to the routing printed on the ticket. So:
- EC379 starts at the central station. Can we use the ticket as is just from there?
No, you have to enter EC379 at Berlin Südkreuz. The ticket is not valid between Berlin Central Station and Berlin Südkreuz.
- If no, can we use the ticket as is for a non "Zugbindung" train (S-Bahn, Regionalbahn) to get from Central to Suedkreuz (instead of starting at Prenzlauer Allee)
Although the routing for local transport is less strict than the routing for long-distance trains, you're still limited to sensible routes without unnecessary detoursThe first trip between Prenzlauer Allee and Südkreuz should be included as a Cityticket (+City on your ticket) on your ticket. FortunatelyAlthough it is possiblewas intended for you to goget from Prenzlauer Allee to Südkreuz via Central Station by taking, it is valid for a local train in Gesundbrunnenpublic transit (thank you @Jan Blumschein), so you can use your ticket on localBus/Tram/U-Bahn/S-Bahn/local trains between Central Station and Südkreuz.) trip from any station inside the Berlin AB zone (*so every station/stop inside the Berlin city limits) to Berlin Südkreuz.
- Do we have to buy a separate ticket from central to Suedkreuz?
No, as long as you use the local trains running between Central Station and Südkreuz. It is not valid in long-distance trains.
Aside from that: the trip between Central Station and Südkreuz takes about five minutes, making it unlikely that your ticket is getting checked that soon. Even if you're getting checked right after Central Station, there is a good chance that the conductor lets you get off the hook with a warning if you're being reasonable about it.
(*) okay, there is a slight caveat: due to construction work there are no local trains running from Gesundbrunnen to Central Station, so you can do this only if your trip is after 13.12.2024.