Landlords of residences must provide a dwelling that is "habitable". What constitutes habitability is defined in law and housing codes, generally at both the state and local level.
If such laws mandate that a dwelling must be provided with AC, then the landlord must provide or repair the AC unit; otherwise, they do not, unless mentioned in the lease contract.
From what I could find, an air conditioning unit is not considered necessary for a dwelling to be legally habitable in TX (as opposed to practically habitable); a garbage disposal is almost certainly not (as one can just throw out food waste like other trash).
Unless your contract bars you from doing so, you may be able to pay for the repairs out of pocket. If it is not your responsibility as a tennent to maintain it, you may be possibly able deduct expenses from your rent (IANAL, I'm from California, talk to a TX lawyer).
Stagnant water is mentioned in the health code, so if it does develop it can be a violation, though it sounds like it is more referencing large bodies of water on the exterior grounds of a property, rather than that what small amount may be in your sink or pipes.
As a general rule, what is required is healthy living conditions for a large swath of the population, including children and the elderly.
If you feel that a working AC unit should be required to make a property habitable, then that isn't a legal problem but a political one: Go forth, citizen, and engage in the democratic process!