Can only married couples apply for financing together?

I would ask around. Who is supposed to become the owner? Someone is off the hook in this story because they are not a borrower but a co-owner.

I would ask around. Who is supposed to become the owner? Someone is clearly getting off easy in this situation, being neither a borrower nor a co-owner

As I wrote, the property (and thus the house) currently belongs to Part 1. That would not change even if the loan is only in Part 1’s name.

As I wrote, the land (and thus also the house) belongs (currently) to Part 1. That would not change even if the loan is only in Part 1's name.

and Part 2 pays indirectly too?

that would be too stupid for me

and does Part 2 pay indirectly?

that would be too stupid for me

The question is a completely different one. Please put the emotional side aside.

This is about one employee and the other being self-employed. Both married. Banks generally penalize self-employment (even as a second borrower). Hence the consideration to take out the loan only with Part 1 (then under better conditions).

If the salary of Part 1 is correct, then that will work. However, we cannot speak for the banks now. You would have to do that yourself.

Banks generally penalize self-employment (even as a second borrower).

That is not true. They just examine more closely because the risk could be higher (but doesn't have to be). The terms do not automatically get worse just because there is an additional co-borrower. What could happen to the bank? The self-employed person drives their business into the ground, and the employee must be able to repay the loan alone. That is no different than if the employee had taken out the loan alone from the start. This is at least how we proceed (we grant tens of thousands of loans to home builders and buyers every year).