Yes.
This all depends on the laws of your state and the terms of the lease. The larger the apartment complex is, the likelier the landlord will have to pay for water, garbage, and sewer.
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.
Unless your rental agreement specifies liability belongs to the landlord, it would be very unusual for the landlord to have any liability.
Your landlord's insurance should take care of it. Legally your landlord is liable as they own the tree and supposedly should have had it checked and trimmed to prevent that.
Yes, it is possible to have a dog in an apartment, but you may need to check with your landlord or apartment complex for any specific rules or restrictions regarding pets.
An apartment complex is almost always private property, with or without a gate or fence, owned by the landlord, or jointly if they are condos. Even if it is owned by the municipality, there may also be a fence or gate and access is restricted to residents and their visitors, although it is not technically "private" property. As part of the subdivision or cluster housing plans, the private apartment complex developer may have also given "dedicated ways" to the municipality, for purpose of access by emergency vehicles and other services. If that is the case, one could argue that such streets must be open to the public, even though they are still "private property". Consider, for example, that a shopping mall, and its parking areas, is also "private property", but open to the public, subject to private rules.
The towing company that tows vehicles for your apartment complex is typically chosen by the property management.
It is the decision of the owner of the property whether he wants to create a smoke free environment for the tenants. It is very difficult for any landlord to be able to enforce a rule against smoking in one's own apartment. As a tenant you have the right to make your own apartment smoke free.
It is important for tenants of an apartment complex to have insurance on there personal property. Theft fire, and flood are among the biggest reasons to have such a policy.
The apartment complex master policy
In NYS, the landlord is not obligated to meet such a request. He may do so at his own option.
The driver