Ask your attorney about your local laws, but because the infestation is passing through common areas I suspect that they can be.
The landlord
It normally depends on how many units are in the complex. If the "complex" is actually a duplex, triplex, or quadplex, then no: it's not an apartment complex and generally the landlord is not responsible for pest control. If there are more than four units then it's an apartment complex, and is subject to more rules: the landlord is normally responsible for the pest control, inter alia. Code Enforcement can help you enforce the laws your by which your landlord must abide.
Yes , but only if the tenant gave a written request asking the landlord to fumigate his premises before occupation or within the first few weeks of occupation.
It depends on two things..which can vary from state to state 1.The landlord tenant laws in your state... 2. The lease agreement. Some States require that the landlord be responsible for all forms of pest control (ie) roaches,rats,mice,bees,bed bugs,ants,spiders,ect ect.. Some States only require the landlord to insure that a rental unit is free from all pests before renting the unit and then any further pest control is the responsibility of the tenant... Lease agreements can also have stipulations for pest control as well according to the laws of the state....(ie) A landlord may be responsible for the control of mice,rats and termites but may not be responsible for roaches,bed bugs and ants... So check your lease and the laws of your state....
Landlord, and their insurance
It depends who threw out the furniture. If the furniture is yours, unless you have a special agreement with your landlord, you must dispose of it. Otherwise you will be responsible for the cost of having it hauled away. However, if the furniture was left by another tenant or by your landlord, your landlord will have to make sure that it is picked up and disposed of (either by the landlord or the tenant who threw it out).
It depends on what it is. If it's something that makes the apartment unlivable, the landlord is responsible to make the repair. If you're looking for something that is considered cosmetic, then you can get permission from the landlord to do it yourself. Try to negotiate with him to pay part of the cost.
Depends on your lease agreement. Usually the landlord is responsible for most repairs but in some instances of insect infestation it will depend on the type of pest and the cause of infestation. Like with some municipalities they state that the tenant could be at fault for the costs. Try asking your landlord if they will cover the cost or even split the cost.
if a landlord bugs my apartment, is that an invasion of privacy?
If the eviction was based on some breach by the tenant, then yes. If it was brought by the landlord simply because he wants the apartment back, or if the landlord lost the case, then no.
Depends on how the lease is written: normally the tenant is responsible for painting interior.
Yes, if it appears that the apartment is abandoned. The general rule is that if your apartment is not occupied for at least 15 days after the rent is past due, then the landlord can reclaim it without taking any legal actions. But there must be evidence that the apartment is abandoned, such as it being empty of any furniture, and that there is no electricity running on electricity is turned off in your name. This is not legally required, but it is a good idea for you to let your landlord know that you will be away from your apartment for the specific number of days you tell him, and you do this by writing.