Bazza: That is very amusing, a landlord would be "happy" to pay for pest control. lol If they are bedbugs, landlord is not liable. If other bugs, you can ask for extermination and allow a couple of tries. If roaches, that comes with the apartment and you have to take care of it. Exterminator cannot do anything.
That depends on your location, what your lease says, what type of building you rent, and what the problem is.
If your lease says you are responsible for pest control, the cost falls on you. If your lease is silent about pest control, your state's landlord/tenant laws prevail.
It depends. There are far to many factors to list. We really cannot answer this without a specific situation.
You should consult your written agreement and if there is something written about it then you should follow it.