Which type of funding should I choose? KfW, BAFA, tax?

Is this still possible at all? I think I read that in addition to the KfW funding, the heating system can no longer be funded through BAFA.

An additional funding was never possible. Only a separate one, if the system was considered in the efficiency house calculation, but the costs were not included in the funding application. Then the funding for the heating system could be applied for separately through BAFA.

Following your hint, I looked it up again (I am currently not working in the KfW / BEG funding area for residential buildings, so I was not up to date. At the moment, I am professionally only doing NWG / energy audits.)

It seems that combining an efficiency house and a BEG individual measure is not possible in the same construction project at the same time. Then the heat pump would have to be applied for and funded through the efficiency house, with a correspondingly lower funding rate, but with low loan interest rates — which only makes sense if equity is not available or can be used elsewhere.

Alternatively, first apply for and get the heat pump funded with surrounding measures through BEG, and afterwards carry out the other work toward the efficiency house 70EE — the accompanying energy consultant will know better about the minimum time intervals involved.

I wanted to use the Schlüter thin-layer system here. I have absolutely no experience with dry screed, nor anyone in my circle of acquaintances who uses it.

Then I would opt for the Schlüter system P/PF, the only one with a 16mm pipe diameter.

Heat pumps require a relatively high flow rate (depending on the model about 12-20 l/min). With 14mm or even 12mm pipes in underfloor heating, you need very short and many heating circuits there or alternatively a very high pump capacity. Schlüter as a system is technically quite overloaded, ultimately there are various suppliers of installation panels and composite pipes. The advantage of the system providers is, of course, that you don't have to rely on the installer's knowledge regarding the design.

Thank you for your feedback, that moves me forward quite a bit.

Since I unfortunately have no idea about the design of the heating loops, a planner would have to take care of that for me.
And currently it is very difficult to find someone you can (also professionally) rely on.
Can a drywall system also work with thermal insulation underneath that can compensate for slight unevennesses in the concrete ceiling?
And which system would you recommend to me here?
(are there any disadvantages compared to "wet" screed?)

Can a drywall system also work with insulation underneath that can compensate for slight unevenness in the concrete ceiling?
And which system would you recommend to me here?
(are there any disadvantages compared to "wet" screed?)

I would fix unevenness in the raw concrete with leveling compound before laying the insulation layer. For larger unevenness (usually approved from 3 cm), you can use load-bearing loose-fill insulation. There are mineral-based options on a clay basis or made of perlite, or also bound loose fills, which are mixed with water and then harden. On top of the loose fill, use as much load-bearing insulation as possible (approved insulations usually have >150 kPa load-bearing capacity at 10% compression), then the underfloor heating setup and screed.

Dry screed has the disadvantage compared to concrete screed of lower buffering capacity, which can be disadvantageous with heat pumps – but not significantly. However, it offers faster response, for example during night setback.

Regarding planning and hydraulics, I would refer again to "Flow 30". This works with dry screed as well as with wet screed, even better with dry screed because it responds faster to overheating.

The most important basis: no system-separating buffer tank / hydraulic separator. This is often installed to ensure the high flow rates required by heat pumps in distribution with small diameter pipes in case of poor hydraulics.

Better use large pipe diameters, 1" or more, for the connection of the heat pump to the distributor, 16 mm+ pipes in the heating circuits, preferably equally long heating circuits (80-100m), small installation distances adapted to the heat load of the rooms, low supply temperatures (30-32°C with NAT).
No balancing of individual heating circuits generally, but only for individual rooms (e.g., bedroom for night setback and storage room) – temperature control via hydraulic and thermal balancing.

In some cases, a buffer tank may be necessary to ensure the required water volume during defrost cycles. Here, do not use a buffer tank functioning as a separator, but a return flow buffer tank.

With dry systems, there are panels for 12.5 cm installation spacing with aluminum coating for heat distribution. Above this, only the dry screed is laid. Some manufacturers even advertise XPS panels without the use of dry screed.

I have already seen or rather felt the variant with dry screed. It works without problems with evenly heated floors. I installed the variant with XPS directly without dry screed in my own bathroom, however in a version without aluminum coating and higher supply temperature, so only XPS panels with milled-in conduit channels, tiled surface. Here the heat output is uneven – I would rather recommend this only with aluminum foil and lower temperature and if the installation height is strictly limited.
I myself will also install the dry system without screed in a building project, but that will take almost 2 years until completion.