Radiometric dating does make assumptions, but they're not as rigid as you've outlined:

1) Time starts at zero for dated material: It's not that time starts at zero, but rather we're measuring the time elapsed since a particular event - usually the last heating or alteration of the material.

2) No contamination of material: Scientists do acknowledge potential contamination, which is why multiple samples and cross-checking with different dating methods are often used.

3) Constant decay rate: This is indeed a fundamental assumption. However, decay rates are derived from…  Read more basic principles of quantum mechanics. To date, no evidence has been found that suggests these rates have changed significantly over the history of the universe.

The Mt. St. Helens example you mentioned is a common misconception. The issue there was misuse of the method; radiometric dating isn't meant for rocks known to be less than a few million years old because the technique isn't precise enough to differentiate between "zero years old" and "a few hundred thousand years old." It's like using a yardstick to measure the thickness of a hair - it's just not the right tool for the job.

As for carbon dating, it's important to note that it's just one method of radiometric dating and is only effective for dating organic material up to about 50,000 years old. It's not used to determine the age of the Earth. Other methods like uranium-lead dating are used for that, which can date materials billions of years old.

The fact that we can still carbon date implies that carbon-14 is still being produced in the atmosphere, not that the Earth is less than 90,000 years old. The production of carbon-14 involves cosmic rays hitting the atmosphere - a process that's been ongoing for much of Earth's history.

Therefore, the presence of carbon-14 doesn't limit the age of the Earth to 90,000 years; rather, it limits the age of the organic material we can date using this method.