"I have found a dead body on my car."

Ignoring matters of ambiguity in natural language (since this does not seem relevant to what you are asking), your sentence could be rephrased as:

$$\text{'I have found a dead body on a car that I own'}$$

where 'a car that I own' is an indefinite description according to Russell's theory of descriptions. The whole sentence may be formalised as follows:

$$\exists x,y : \mathrm{car}(x) \wedge \mathrm{body}(y) \wedge \mathrm{dead}(y) \wedge \mathrm{foundOnTopOf}(i,y,x) \wedge \mathrm{owns}(i,x)$$

If you don't own a car, then the statement is false, since you are in part asserting that there exists a car that is yours.

There is no implication in your statement. A natural translation into first-order logic would be something like "There exist a car and a person such that: the car is mine and the person is dead and the person is on top of the car."

If you do not own a car, then the statement is false since "the car is mine" is always false.

If you do not own any cars, the following statement will be true: "I have found a dead body on each of my cars."