Investigation of heating costs before property acquisition?, Check of quote

Does every house have to have an energy certificate and if not, is it of course that I as a buyer have to bear the costs for it?
Yes, your seller must present this energy certificate to you upon request!
§ 16 Abs. 2 Energy Saving Ordinance
Commissioning and payment are of course the seller’s responsibility!
Cf. Wiki->Energieausweis.

Without delay means here without delay - and not that he only commissions one when the interested party asks for it! This already puts him in the area of an administrative offense according to §27!

...(Found out that with pellets compared to oil you save about 12,000 euros after 20 years if you install both systems completely new).

I wouldn’t be so sure about that, but it depends on who sponsored it. ;-)

...Otherwise I would be buying two heating systems since the oil heating is already there.

Right! Actually nonsensical. However, the purchase price should meet today’s necessities and demands. An outdated heating system with poor JNG certainly justifies price reductions.

....Does every house have to have an energy certificate, and if not, is it obvious that I as the buyer have to cover the costs?

There is a legal entitlement. The seller must cover the costs.
However, the energy certificate only permits a qualitative assessment or a qualitative comparison of different properties.
Actual expected consumption costs cannot be derived from it. This is also not intended by the legislator.
If you want to know the actually expected consumption costs, additional effort is required.
By the way, Micha&Dany described the overall connections very well!

Best regards.