The short answer is yes, duct cleaning, done right by a reputable company will help with the smell. It is likely though the smell is coming from another source, ie the crawl space. It could also be that another rodent is dead in the duct work. A visual inspection would be in order to check for any breach in the duct system. A duct pressure test would be able to tell you how much air is leaking from the system over all. A hole large enough for an animal to enter would loses a great deal of air, over 300 CFM. It would be worth looking into having this test done and it should cost around $200. Cleaning the air handler and evaporator coil will greatly improve the performance of the air conditioner. Dirt and dust on the evaporator acts as insulation and reduces the amount of cooling done be the coil. Any quality duct cleaning should include cleaning the air handler and coil, and the tech should allow you to watch him perform the work as it is done.
My Carrier furnace occasionally (one-two times a day) does not start. It shows error code 15 ("BLOWER MOTOR LOCKOUT -This status code indicates the blower failed to reach 250 RPM or the blower failed to communicate to the variable speed furnace control within 30 seconds after being turned ON in two successive heating cycles."). However, when I turn off and on the furnace power switch it starts and runs fine for a while. Sometimes it runs fine all day, but does not start only once in the evening or in the night. It is 13 years old I was thinking to replace whole thing if repair cost is too high. But I am not ready to pay $10,000. 
I hope that those customers you have with breathing problems don't follow your advice too closely. You are doing them a disservice. Dirt in your ducts does not always stay there, depending on what the contaminants are. I bet you tell your customers not to worry about mold either- just to kill it with bleach. Educate yourself before opening your mouth- You may kill someone someday with your opinion.
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My Carrier furnace occasionally (one-two times a day) does not start. It shows error code 15 ("BLOWER MOTOR LOCKOUT -This status code indicates the blower failed to reach 250 RPM or the blower failed to communicate to the variable speed furnace control within 30 seconds after being turned ON in two successive heating cycles."). However, when I turn off and on the furnace power switch it starts and runs fine for a while. Sometimes it runs fine all day, but does not start only once in the evening or in the night. It is 13 years old I was thinking to replace whole thing if repair cost is too high. But I am not ready to pay $10,000. 
Compressor -- Compressors are outdoor components in your system. The compressor is the pump that circulates the refrigerants through the air-conditioner. You can hear it when it's running so you'll know if it's working just by listening. If it starts getting louder, your compressor is about to fail. If it makes no sound when it should be on, it has already failed. Compressors fail for a number of reasons. Most often they fail due to strain from another failed part such as the fan motor. Electrical storms can also damage compressors. If the sound from the compressor gets louder or if you see a decrease in performance, you should have your compressor checked. A failed compressor will not heat or cool your house.
I am NADCA Certified Air Systems Cleaning Specialist. Over the years I have learned on simple answer to this questions: there air duct cleaners and there are pretenders. There will be no benefit at all if you use on the very cheap pretender because they will not really clean anything. On the other hand if you call a qualified air duct cleaner with a NADCA Certification you will, like 99% of my customers, report improvement with your respiratory problems, you will see less dust and the job will pay for itself with energy savings and repair prevention.

I to had an odd layer of white'ish dust building up especially during seasonal periods when the HVAC ran a a lot. I carefully swept up the fine dust into a small pile and had it analyzed. Turns out it was very fine fiberglass powder dust. It turns out it is the insulation form the attic! (My attic has blown in insulation). Turns out if you (I) had small gaps around my HVAC registers in the ceiling whee the airs blows into the room. What occurred in my case was that the exiting air at that seam creates a vacuum effect at the grate vent cover and draws and then blows fine insulation dust from the attic into the house. Removing each register grate and sealing each tin outlet gap with calking between the dry wall ceiling and tin fixed the issue. A couple were so bad with bowed gaps I had to screw down the bowed tin to the attaching stud better. Checking the very same glass top desk that catches and shows any dust accumulation now shows only very small amounts of normal house dust build up over several weeks. A big, big improvement. You might have a look by simply removing a couple vent covers yourself.
I've been in business in Florida for 25 years and purchased a $12,000 duct cleaning system 4 years ago. In the last four years I've cleaned 5 duct systems charging around $400 each. Not a very good return on investment. What I have been doing most the time is I get a call by someone panicked because they called some low priced company to inspect their system and were told they have dangerous black mold. Then they call me to have a look. Most the time they don't have mold, ducts look very clean. However the coils are dirty and the inside of the air handler cabinet needs cleaning. Keep in mind not every system can handle supper high efficiency filters! These filters can block air flow quite a bit right out of the package. Many older systems can't handle pleated filters due to increased static pressure. "Back rooms with low air flow". I have taken many hours of classroom study on indoor air quality. Never got any certification because I don't want to pay another organization dues yearly. I think the problem with duct / system cleaning is most people are suckers for low priced scams that are rip- offs . People call me all the time and ask how much? When I tell them average $400 they hang up. I know I do the job correctly and honestly, and they are going to get racked over the coals by the scam artists by calling $79 whole house duct cleaning guys. People need to check internet, angies list and BBB reviews. But keep in mind even reviews need to be sifted through your rational person filter.
Quality and price should come hand in hand, a too-good-to-be-true price for an efficient air duct cleaning will not give you the quality that you wanted. To make sure you are getting your money’s worth, check reviews of air duct cleaning companies, certifications from industry since this work requires continuous learning due to new techniques and research breakthrough from time to time. My mom is located in Arizona and they trust J&M Restoration air duct cleaning company, they have an A+ BBB rating, they used EPA approved sanitizing agents, their technicians are NADCA certified, and the company is certified by the Institute of Inspection, cleaning and Restoration. Make sure to shop around and get written estimates first.
Chemical biocides are regulated by EPA under Federal pesticide law. A product must be registered by EPA for a specific use before it can be legally used for that purpose. The specific use(s) must appear on the pesticide (e.g., biocide) label, along with other important information. It is a violation of federal law to use a pesticide product in any manner inconsistent with the label directions.
When central air conditioning service fails during a heat spell, you may have to wait days for an HVAC repair technician or an ac contractor to show up, and you'll probably pay at least several hundred for the repair. But if you're comfortable working around electricity and are willing to spend about $50 on parts, you can probably repair your air conditioning service yourself in about two hours and save about $225 on parts markup and labor.
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