Einstein's theory proved that most properties of a body are relative to something and are not absolute. I am still wondering if physicists still think that there is a way that Newton's absolute space and time still exist? If they do, how are the ways that we prove their existence? and how do they exert their influence (effects)?
Generally, no, Newton's concept is now seen as a limit case of Relativity - you only get (exact) Newtonian mechanics when the velocity and masses of all objects involved reach zero. There have been various attempts to tweak Newtonian dynamics to make it match reality (known as MoND, Modified Newtonian Dynamics), but without success, there's always some observation that doesn't match.
Not that Newtonian dynamics isn't useful, it's a much simpler way to calculate as long as you're dealing with a situation where the deviation from Relativistic dynamics is negligible, but in the strictest sense it is always "wrong" to some extent.
Spacetime is a consequence of Einstein's false constant-speed-of-light postulate:
"Special relativity is based on the observation that the speed of light is always the same, independently of who measures it, or how fast the source of the light is moving with respect to the observer. Einstein demonstrated that as an immediate consequence, space and time can no longer be independent, but should rather be considered a new joint entity called "spacetime." http://community.bowdoin.edu/news/2015/04/professor-baumgarte-describes-100-years-of-gravity/
So spacetime doesn't exist, space and time do. Einsteinians know that:
Nima Arkani-Hamed (06:09): "Almost all of us believe that space-time doesn't really exist, space-time is doomed and has to be replaced by some more primitive building blocks." https://www.youtube.com/watch?v=U47kyV4TMnE
"Rethinking Einstein: The end of space-time. It was a speech that changed the way we think of space and time. The year was 1908, and the German mathematician Hermann Minkowski had been trying to make sense of Albert Einstein's hot new idea - what we now know as special relativity - describing how things shrink as they move faster and time becomes distorted. "Henceforth space by itself and time by itself are doomed to fade into the mere shadows," Minkowski proclaimed, "and only a union of the two will preserve an independent reality." And so space-time - the malleable fabric whose geometry can be changed by the gravity of stars, planets and matter - was born. It is a concept that has served us well, but if physicist Petr Horava is right, it may be no more than a mirage. Horava, who is at the University of California, Berkeley, wants to rip this fabric apart and set time and space free from one another in order to come up with a unified theory that reconciles the disparate worlds of quantum mechanics and gravity - one the most pressing challenges to modern physics." https://www.newscientist.com/article/mg20727721-200-rethinking-einstein-the-end-of-space-time/