COFUND: An entity with a successful COFUND project in Horizon 2020 would like to submit a new proposal in Horizon Europe. Will it be checked if the previous proposal and the future proposal are under the same Framework Programme (Horizon 2020 versus Horizon Europe)?

There are no such restrictions. If the proposal is excellent, it will get funding, no matter if it is within the same Framework Programme or not. There are examples of organisations with consecutive programmes within the same Framework Programme.

COFUND: An entity with a successful COFUND project in Horizon 2020 would like to submit a new proposal in Horizon Europe. Is it true that if an organisation succeeded in a Postdoctoral Programme, it is advised to opt for a Doctoral Programme in case of a new submission? And vice versa?

There are no such restrictions. If the proposal is excellent, it will get funding, no matter if it is only a Doctoral Programme or a Fellowship Programme. There are examples of organisations with consecutive programmes.