I have an early 1950's second home in Cleveland area. It has knob and tube wiring, which as mentioned would generally be safe for the long runs even without insulation. That is, until it gets to an outlet. Does your house/apartment include an attic or basement? Is there evidence of rodents there?
Flickering lights in the Cleveland home have been due to:
1) Loose connections in the electrical panels (there are 4 panels from various updates and additions);
2) A loose phase wire at the utility pole; and
3) Low amperage capacity, e.g., the dining room lights flicker when the 110V washing machine runs, but not the 220V dryer.
Notably absent, no rodent problems.
As for mice, are you sure they are not rats? In large Eastern cities with row houses (e.g., Baltimore, NYC, and Boston), rats can be a problem and are very hard to trap. I am familiar with row houses in Baltimore, and they all were separated by brick walls, even ones near the harbor and dating back to the Clipper Ships. See:
http://www.tonerarch.com/blog/2013/6/13/your-house-and-your-facade-a-separation-agreement.htmlAs for responsibility for wiring, that is the apartment owner. You would also need the owner's permission in theory to make any holes in the walls. The walls are probably originally lath and plaster. Even old newspapers and horse hair was used as early coats. It is easy to repair small holes in them. They may have a drywall covering as well.
I don't know who is responsible for rodent control. In theory, if you were to call an exterminator, that would be the owner's responsibility to pay, as non-payment could result in a lien on the property. In practice, renters are pretty much responsible. Many row houses are individually owned. In that case, it is best just to talk with the neighbors.