There is no compensation due here and even the offer of the day room was simply a customer courtesy.

OP chose to delay himself and wait for his checked luggage when there was no reason to do so. First, why ever check a bag which does not need to be checked. It is always easier and cuts the risk of the bag going missing. Second, the bag would have been delivered to OP's ultimate location even though there may not be direct service to HEL. It happens thousands of times/day, if not more, around the world.

Generally luggage can make the transfer if the passenger can. That is what the ground agent at SIN and the FA (stewardesses haven't existed for 35 years!) effectively told OP. They have no way of knowing, to the minute, what will actually transpire on arrival and gave the stock answer they are trained to give.

OP apparently purchased a separate ticket for his onward journey. It was not QR's responsibility to arrange airport transfers from him from FRA.

If OP is upset, he might drop QR a short (3-4 short sentences, not the above which is way too long) and he might see some minor customer service gesture, but compensation just is not in the cards.