Pull up the threshold and look. You should be able to detect how many layers there are present. I'm not clear on why you feel you need to remove the vinyl. You should be able to buy hardwood flooring which is about the same thickness as the laminate you're replacing. If your goal is to do away with the threshold, however, you are going to have to be very specific about the thickness of whatever you buy to replace it with, to the point of possibly compromising your otherwise-ideal choice.

much like pain is a symptom of a medical issue that needs addressed, dirt/dust/debris gathering in corners is a symtom to the condition that the flooring area needs cleaned.

If you want a flush transition, and the underlying surfaces are flush, you only have 2 choices ...

1. Remove the laminate, then install new flooring in both areas so the transition remains flush. DO NOT install another laminate as the doorways will require the very transition you now despise because of expansion and contraction needs of the product.
2. Hire an Asbestos Abatement Contractor to remove your resilient floor system. Then remove/add underlayment/subfloor as needed until your new flooring choice becomes flush with the existing floor you are trying to match height to. Again, stay away from any laminate flooring choices, because they require a transition/reducer/molding that will sit above finish floor height to allow for expansion and contraction of the laminate flooring system.