Previous tenant did some shoddy DIY, should we be concerned?
That was not previous tenant work. That was shoddy handyman work over a long period of time. I'd venture to bet it was the LL himself trying to save money. 

Bottom line is: You CANNOT do ANY electrical work in a place that you are renting (if that's what you are asking). Your only recourse is to contact the LL and tell him about it and ask him to make the repairs corrections, or have it done by someone. Than again, from the looks of it I bet he already knows and does not care. I'm curious, did you not see any of this before you signed the lease??

The wiring shown is phone wire and poses no safety hazard. Unless of course you are holding bare wires when the phone rings. Then you'll get a small shock.