I have my reservation about adding to this discussion because the discussion would be deep and answers would be shallow. Your point that gravity for a small particle can extends infinitely made me feel I should poke in... (Because observed fact is more interesting than that...)
Physicist really don't have all the answers. Our brain equipped us well to deal with the everyday thing, like "lion chasing us". Our brain also is well developed evaluations of information our brain received, such as catching a ball thrown at us. We could catch a falling apple long before we understood gravity, parabolic equations, or laws of motions. We can evaluated the intercept point almost like it is done by magic. That is because our brains are developed to do that. Physics now dives into things that our naturally developed brain doesn't really equip us to deal with -- because it was not essential to our survival, our brain doesn't need to develop ways to deal with it for our species to survive to present day.
Physicist develop models to deal with it. A model can help explain it, but it doesn't mean that "it is actually that way". Further, the model could be only applicable under limited condition. Newtonian mechanics for example, the model it describes works well for magnitudes of properties we typically deal with. Such as an apple falling from the tree. The magnitude of it mass and speed are "typical of what we can deal with" because that was what we were able to see, measure, and test. The model so developed are applicable within those limitations but may not function beyond that. When that apple is traveling near the speed of light, that is not of the magnitude we are accustomed - it has not been seen or measured or experimented. So new models may have to be developed. As it turn out, F may still equal MA, but M doesn't stay the same -- the apple becomes more massive.
I would think gravity doesn't extend infinitely. Gravity should be quantized and doesn't exist below a certain point. But, observation is more interesting than that...
Why quantized? There is a limit to how small "length" can be. "Length" is not meaningful below a Planck Length (1.6x10-35 meter). How long does it take for a particle of light to travel it? A Plank Time. Below that (shorter period than that) time is not meaningful. So if the gravitational force on this particle (A) of a far enough away particle (B) can only move A&B together almost one Planck Length but it take many times more than projected live of the universe, so is it moving at all? Force equation is F=MA, A is acceleration, Time and Length are variables in acceleration. Thus, if Time and Length are quantized, F must be quantized as well.
Observed fact got in the way. That a particle can have gravity and that gravity is proportional to the inverse of R2 is another model that fails when beyond it's limit. As R (ie: distance) increase to galactic scale, it is unclear that it follows the classical equation F=G*M1*M2/R2. In fact, the observed speed of the outer stars of galaxies are traveling faster than can be explained with above stated equation, gravity seem bigger than the equation says what it should be -- hence "dark matter" (missing mass you can't see) is proposed as a model to explain this observed phenomenon and movements of galaxies. Dark matter is not the only model. A competing model is "Modify Gravity" theory that gravity doesn't decrease by R2 but instead it changes some other ways. Some physicist would consider them both (dark matter / modify gravity) the same thing. This "same thing or not" is instructive. Whether you modify gravity by changing the equation or you modify gravity by inserted some free-variable (dark matter) is irrelevant - either way you are changing the model. How this will settle is at yet unclear. So, may be it does extend infinitely, or may be it is quantized that below a certain minimum level it cannot exit.
If you study more Physics, the world will be more interesting to you. But you likely will die someday not understanding the world you are in still. There is a lot more than a few life times of learning and discoveries yet to be done.
Go study, read more, learn more. The universe is one interesting place. I am not sure us humans will exist long enough to fully understand how the universe works, but it would be an interesting journey.
EDIT:
Moved paragraph "Why Quantized" up (was in the wrong place) and a typo correction. Removed an unnecessary (and repeated) comment about human's ability to understand.