Is this cosmological scenario possible? Is it possible that the universe is infinitely large and contains an infinite amount of mass that is distributed in such a way that gravitational force is never infinite? If so, is it possible that the infinite amount of mass that is outside of our light cone could affect us with gravitational force?
 A: Is it possible that the universe is infinitely large and contains an infinite amount of mass that is distributed in such a way that gravitational force is never infinite?
Yes, in the sense that it is possible that for each mass, the total of all the forces on it is finite. To avoid an infinite force we should avoid point sized masses, so we'll use some rigid balls of mass 1 gram and all of radius 1cm.  Then we can place one at the origin. Then 4 more in a square around it at the four corners, with a length of 2m, with the first ball at the center.  The total of the forces on the center ball is $2G$ $g^2/m^2$ (and the total force is zero since forces add like vectors).  Next we place 4 more, but on a square of 2 times the length (also centered on the first ball).  The total of the forces due to those is 1/4 as much, so $G/2$ $g^2/m^2$.  Do it again, four more masses, on a square of 2 times the length of the last one, so get 1/4 the total of the forces.
So we have the total of the forces from all the squares on the center ball is 
$$ 2G g^2/m^2 (1+1/4+1/16+1/64 + \dots) = 2G (4/3) g^2/m^2 < \infty$$
And that's the mass that is closest to the most, so the total of the forces on any other ball is lower.
Basically when you through the word infinite around its hard to conclude much of anything that must happen, because there is a lot of leeway.
If so, is it possible that the infinite amount of mass that is outside of our light cone could affect us with gravitational force?
We don't know whether there is an infinite amount of mass outside our past light cone.  But something currently outside our past light cone can have an older version of itself that was inside the cone, and that can affect us.  Just like someone can write you a letter before they die but it might not get to you until after they are dead, you can still read the letter and be affected by their earlier thoughts that happened early enough to affect you now.
And again, infinite or finite wasn't the issue here for this second question.
