I'll try to respond to your statements one-by-one, I think you're right overall, but confused on some fine details.
1) yes, your perception of time is related to your speed (imagine you have a watch on), the faster you travel, the slower you experience time (your watch slows and things around you seem to be going faster because your actually going slower); in theory, at the extreme of the speed of light (the maximum speed allowed in the universe) you will no-longer experience time (your watch stops, you may move a great distance, but to you it'll have taken no time at all). The same is true if you're in a gravitational field, the stronger the field, the slower your watch will run. (and in the extreme, if you hit the event horizon of a black hole, in theory you will no-longer experience time, your watch stops, again)
2) Mainly, according to the theory of relativity, no massive object (i.e. object with any mass, from a heavy planet to a puni particle) could ever reach the speed of light, it would require an infinite amount of energy as the mass would tend towards infinite.
I tried to answer further, but things get hard when you ignore one fundamental law and assume things could work...
3) Firstly, you've made a bit of an inaccurate statement saying “the gravity of the car would not only pull on the gravity of the Earth...”; in the case of falling bodies you should say that the gravity of the Earth pulls the Car... Gravity pulling gravity sounds confusing.
But yes, you're right, the ether model was disproved; however Newton's theories can be applied if you set any arbitrary reference frame; though in relativistic extremes, his theories don't work.
In relativity, the very concept is that any “inertial frame of reference” (which is a complicated concept) is a good frame of reference; so for example, if I wanted to calculate what happens to things around me as I sit in a moving spacecraft (moving in a straight line at a constant velocity), then I could use myself as a stationary point in my reference frame, and the maths will workout for everything around me.
Equally, it's just as true that for another spacecraft (moving in a straight line at a constant velocity, even if it doesn't match mine) can decide that they're the stationary point in their reference frame, and the maths still all works out.
(it doesn't matter which inertial reference frame I use, both work)
In the long run tho, Newtonian dynamics are wrong, but only in extreme cases (moving near the speed of light, in strong gravitational fields...); but as a general rule you would never notice the difference. Of the 8 planets in the solar system (no Pluto, sorry), you can almost exactly calculate the motions of 7 of them with Newton's laws (Mercury is so close to the sun it causes issues), and the last is only inaccurate by a few percent.
I'd love Universe Sandbox to support relativistic physics, but considering you need a degree in maths to even touch upon it and it'd definitely eat my laptop for breakfast (the maths of relativity is notoriously hard, unlike Newtonian dynamics)...
I'm OK with Newtonian dynamics for now...
In-fact, most models of the Universe we run on supercomputers use newtonian dynamics, if we didn't then those suer computers wouldn't be anywhere near powerful enough.
I hope this helps.
Please, if anybody notices a mistake with my post, then tell me... Thanks