Is it permissible to buy a house for $110,0—$70,0 cash and $40,0 paid to the bank—knowing that the seller took out a loan on this house, that we have $90,0 available, and that the house needs an additional $25,0 to complete construction and finishing?

If the intention is that the seller will sell the house for one hundred and ten (seventy thousand paid directly to the seller and forty thousand to the creditor bank), then there is no sharia impediment to that, as long as it does not lead to a prohibited outcome.