I did not speak of rent-free. It is generally very difficult to find a fair solution when the house is supposed to belong to only one partner. Therefore, I am not in favor of these arrangements because one person is always disadvantaged. Either a partnership is new... then renting would be great, or a partnership is so stable that such a project should be undertaken together. Marry first, then buy and pay off jointly is actually the cleanest solution. Marriage is recommended because from that point on the community of accrued gains applies. This appropriately takes into account the gift from the parents through the initial assets. In case of separation, 50% of the accrued gains are then paid out. Very fair.
If she pays 50% of the installment, she should of course also acquire 50% of the ownership claims arising from the repayment.
Whether and how this can be secured civil-law wise without land register entries, I do not know.
Yes, with a marriage. Whoever has the ownership shares must pay out the other in case of separation. This happens automatically.