Mice and bugs are everywhere. And there are many many more of them than there are of us.
They are common to many residences if they are not kept out.
Keeping them out is the first order of business. I don't know what your landlord is required to do but what you can ask them to do is fill any holes in your apartment and hire professional extermination services on a repeat basis until the problem is solved.
You can also look for holes to plug (around radiators and baseboards are a good start), and also consider getting something to block 'drafts' under the door to the hallway or outside. those draft blockers often keep pests out. There are some over the counter pest control sprays that are useful for bugs that you can administer yourself as well. Ask at home depot or the hardware store for the good stuff.
Making your home less attractive to mice can be helpful. Food that comes in cardboard boxes should be transered to rubbermaid type tubs, or put in the fridge. Don't leave food on the counter, or in drawers in easily accessible packaging. If you see something they've been snacking on, or 'droppings' concentrated in one place do your best to remove temptation. (this is not a comment on yhour housekeeping, most of us don't double wrap our cereal until uninvited guests force our hands
As for the bugs in the sink its hard to know what to do about them without knowing what they are. Many insects are attracted to moisture (silverfish come to mind) so drying the sink after use may be helpful.
Poisioning mice is a very bad idea because you have no control over where they die. When they die in the walls it takes a while for the smell to die down. Try regular traps or, if all else fails, a cat.