I am a professional floor covering installer with 30 years of experience. I have 18 employees, and do all commercial work. I am a Forbo Master Mechanic. And the list goes on.
Almost all manufacturers state that you must install any new floor on a properly prepared substrate directly for your new floor to be covered under any warranty.
In California it is against the law to install a 3rd layer of flooring.
The problem is you don't know where your weak link exists, and your new floor will most likely fail. All new floors will create stress in time, and pretty quick too. That being said you will have quite bit of stress in the form of shear. The adhesive you will most likely use will have enough strength to hold against shear, but at least one of the layers below will not. So my professional advice is don't do it.
As a professional this is a job I would walk away from, If I could not remove the existing material because a customer wanted to save money.
As far as Asbestos being present in any of the layers is concerned, there is a high likelihood. All 9 inch resilient tile has Asbestos in it. Most resilient sheet goods do not. No Linoleum has ever been manufactured with asbestos, and the two main ingredients in linoleum is actually Linseed oil, and Cork flower. That being said though, many of the mastics used prior to 1984 a percentage of chrysolite Asbestos material. Black mastic is a big red flag and needs to be sampled and tested from any structure built before 1984. This information is in the Code of Federal Regulations 1910. and it is also true if you are removing any painted surface i.e. baseboards due to the presence of Lead in older paint.