Not knowing where you are residing from your inquiry, different solutions to your concerns may be customary in your area. In the Orange County, CA area, it is not typical of a tenant to conduct a professional inspection on a home they are leasing (may actually be a good idea though). Regardless, if the outside deck is determined to be unsafe due to termite activity and/or damage, I would think the landlord would want to remedy the safety hazard. Mildew or mold discovered in the laundry room may have been a pre-existing condition, as it doesn't necessarily materialize overnight.
The important treatment of any mildew or mold concern is to remedy the source of the moisture to curtail further damage and prevent a future reoccurrence. Again, I would think a conscience landlord would want to address this issue promptly...it only grows worse. Garage A/C units are uncommon in our area, and may not be high on a landlord's "to do" list, unless there was an agreement to include this amenity (I.e., storage of items that needed to be kept at a certain temperature, etc.).