I have asked the plumber responsible for the work done whether it is normal for the laundry tub (now requires an electronic pump to drain) has water backing into it when it is rested. Approximately two inches fills the tub even after it is totally drained. Knowing very little about this stuff I hap hazard the guess that it was missing one of those "belltraps" under the sink? or is it normal as he claims? Also - the now "no longer used" pipe that was previously used to drain both the laundry tub and machine (that has been cut down as shown in the picture) should that be plugged up or???? I suppose I should not question but this plumber lad is the same fella who has reversed the hot and cold taps in several areas of the house - including the water for the laundry tub so you can appreciate my frustration in not knowing whether he is just pacifying me with "that's the way it is" - Curious - the second photo shows how the water is now being routed out - is this creative or normal? and yes... that too leaks.
Leaking plumbing is not normal, period!Water standing in tubs or appliances is not normal either. Faucets that are backwards are not normal. None of this is even acceptable.
The pumping of water uphill out of a low basement could be acceptable, if it did not create the above problems. By the way, a trap under the sink is not necessary in this case because the pump on the floor is essentially the trap. For this to not leave water standing in the sink you would probably need a quality back flow valve to prevent the water in the pipe above the pump from flowing backwards as soon as the pump stops. If the pipe that goes into the floor is connected to the drain system, even if it is broken, it is unsanitary for it to be left open. And a broken pipe in the floor could still be receiving water from the other end and preparing future problems under the concrete. I don't know what province you live in, but in most provinces, a visit by a city inspector should raise enough health questions (dripping drainage water and mould potential, just look at the wallboard behind the sink) to force the landlord to do a better job -- and probably with a plumber that knows the hot from the cold.