Absent unusual conditions, described in the details of the statute, your landlord is required to provide a working deadbolt on entry doors.
Quote Quoting California Civil Code, Sec. 1941.3.
(a) On and after July 1, 1998, the landlord, or his or her agent, of a building intended for human habitation shall do all of the following:

(1) Install and maintain an operable dead bolt lock on each main swinging entry door of a dwelling unit. The dead bolt lock shall be installed in conformance with the manufacturer's specifications and shall comply with applicable state and local codes including, but not limited to, those provisions relating to fire and life safety and accessibility for the disabled. When in the locked position, the bolt shall extend a minimum of 13/16 of an inch in length beyond the strike edge of the door and protrude into the doorjamb.

This section shall not apply to horizontal sliding doors. Existing dead bolts of at least one-half inch in length shall satisfy the requirements of this section. Existing locks with a thumb-turn deadlock that have a strike plate attached to the doorjamb and a latch bolt that is held in a vertical position by a guard bolt, a plunger, or an auxiliary mechanism shall also satisfy the requirements of this section. These locks, however, shall be replaced with a dead bolt at least 13/16 of an inch in length the first time after July 1, 1998, that the lock requires repair or replacement.

Existing doors which cannot be equipped with dead bolt locks shall satisfy the requirements of this section if the door is equipped with a metal strap affixed horizontally across the midsection of the door with a dead bolt which extends 13/16 of an inch in length beyond the strike edge of the door and protrudes into the doorjamb. Locks and security devices other than those described herein which are inspected and approved by an appropriate state or local government agency as providing adequate security shall satisfy the requirements of this section.
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If there is a smoking area that is actually in a building, then it would be a stretch to call it a non-smoking building. Is the smoke reaching your unit from within the building, or through outside windows? Is the garage actually physically part of the building beneath your unit?

Are there designated play areas in the complex? Is this complex subject to special anti-loitering ordinances relating to "loitering" due to the crime or drug activity in the area that apply in the area where your child was playing?

It's not up to you to decide what other people need in order to be accommodated, just as it is not up to others to dictate to you what is or is not reasonable accommodation. Their rights need to be balanced with yours and your daughter's. Did they actually tell you that she cannot use the common room, or did they simply say that she can't walk in circles around the room? What did they tell you when you asked for a place where your child could walk? If the office that used to be a gym is in fact unused, how about proposing that as a location for indoor exercise?