This will be governed by NH Housing Standards, with perhaps stricter requirements at city level. It does look like you should have recourse for the broken heater, if the heat coming from other rooms doesn't keep the room with the broken heater at 65 degrees. (Practically speaking... the broken heater, while annoying, may not be a problem. Heaters in other rooms will provide heat, and you will absorb heat from other apartments. This is still something I wouldn't want to deal with, though.) It's not clear whether lack of a railing, rather than existence of a structurally-unsound one, is covered. Still, as a tenant, you should do your due diligence. You had to have at least known, moving in, that you needed railings for safety reasons, and that the railing was non-existent. Why not move on, or ask for it upfront and move on if denied? Why move in and then make demands for things you should have known about when you signed your lease?
Section 48-A:14 Minimum Standards Established.
48-A:14 Minimum Standards Established. – No landlord, as defined by RSA 540-A:1, I, renting or leasing a residential dwelling in a municipality which has not adopted ordinances, codes or bylaws pursuant to this chapter shall maintain those rented premises in a condition in which:
I. The premises are infested by insects and rodents where the landlord is not conducting a periodic inspection and eradication program;
[Paragraph I-a effective January 1, 2014.]
I-a. The premises are infested by bed bugs and the landlord is not conducting a periodic inspection and remediation program. In this paragraph "remediation'' means action taken by the landlord that substantially reduces the presence of bed bugs in a dwelling unit for a period of at least 60 days;
II. There is defective internal plumbing or a back-up of sewage caused by a faulty septic or sewage system;
III. There are exposed wires, improper connectors, defective switches or outlets or other conditions which create a danger of electrical shock or fire;
IV. The roof or walls leak consistently;
V. The plaster is falling or has fallen from the walls or ceilings;
VI. The floors, walls or ceilings contain substantial holes that seriously reduce their function or render them dangerous to the inhabitants;
VII. The porches, stairs or railings are not structurally sound;
VIII. There is an accumulation of garbage or rubbish in common areas resulting from the failure of the landlord to remove or provide a sufficient number of receptacles for storage prior to removal unless the tenant has agreed to be responsible for removal under the rental agreement and the landlord has removed all garbage at the beginning of the tenancy;
IX. There is an inadequate supply of water or whatever equipment that is available to heat water is not properly operating;
X. There are leaks in any gas lines or leaks or defective pilot lights in any appliances furnished by the landlord; or
XI. The premises do not have heating facilities that are properly installed, safely maintained and in good working condition, or are not capable of safely and adequately heating all habitable rooms, bathrooms and toilet rooms located therein, to a temperature of at least an average of 65 degrees F.; or, when the landlord supplies heat in consideration for the rent, the premises are not actually maintained at a minimum average room temperature of 65 degrees F. in all habitable rooms.