Can someone tell me if this job might entail chemical or carbon monoxide exposure?

My father operates a rigging business, that routinely hauls heavy machinery, printing presses, etc. around the country to various corporations and companies. As such, he owns a slew of eighteen wheelers, fork lifts, and a warehouse inhabited heavily by boxed up printing presses. Almost constantly, a fork lift is toating around the verious machinery in the warehouse, eighteen wheelers are driving around the building and occasionally inside for maintenance, etc. I would be working in the office area, though the air from the warehouse still gets in it. The fork lifts are all routinely checked to make sure they're not putting off dangerous carbon monoxide levels. If I worked in the office, could I be having any exposure to hazardous fumes or chemicals, or any dioxins or things? Could I lose a few thousand brain cells? It's an old warehouse building, with old and weak ventilation. The "office" area is basically an enclosed set of rooms that intrude into the warehouse itself, sharing air.

In addition, there is grease from the machinery everywhere. Is the grease innocuous, or could it contain heavy metals like lead, arsenic, mercury, etc?

Lastly, a laser printer is used often in the office. I have heard that these emit dangerous fumes that represent a danger quantatively equivalent with the hazard poised by cigarette smoking. Should this be a concern?

Suggest to have an air meter installed to monitor any toxic gasses that may be at dangerous levels. They are not that expensive or difficult to install. They will alarm when levels reach a hazardous point. I would say you have the right idea to be concerned. Metering the air will lower the insurance rate to the facility and will show concern for the employees health, and cutting health care costs by lowering the cancer risk to all.

It would not be ethical to concern yourself only with your own breathing space and not that in the rest of the warehouse. Either clean up the whole space or live with the results in your office space.

Now this is a legal concern. If you have cause to be concerned for your own air quality, and fail to take care of the air others are subjected to, it makes you negligently responsible.

The situations you describe are not extremely negligent, but applying filtration for the air, vacuuming up dust from surfaces so that it is not disturbed into the air would be minimal care.
Anyone servicing filtres or handling dust from the vacuum should be wearing dust masks, washing hands, laundering clothing, to minimize their contamination.

Keeping floors vacuumed regularly is your biggest payback if you have any dust.

Large amounts of dust can be explosively combustible. Be careful of vacuuming with a spark generating cleaner.