if there was a sigularity, of inifinate density, shouldn't that mean infinate gravity? So shouldn't everything be attracted with an inifnate force, and shouldn't we accelerate to an infinate speed towards the singularity?

Often articles talk of the beginning of big-bang as a point of infinite gravity. But, it is just an approximation. It could not be infinite in the strict sense of the term "infinite" because Quantum Physics prevents it.

Could it be that if black holes are made, they should eventually attract everything towards them, including each other, and so eventually the universe would be nothing but one black hole?

Only things which are inside the event horizon of a black hole will be pulled to the centre of the black hole. But, things outside the event horizon need not have this fate.
Moreover, the universe is really a black hole.

but if time fails to exist in a black hole... and all matter is eventually sucked into one... then would time stand still?

Let us see what general theory of relativity says about time inside a black hole. Let us assume that you are inside a black hole and just assume, for the sake of discussion, that you are surviving. I am outside the black hole. According to me (who is outside the black hole) no changes are happening inside the black hole. So, I will feel as if time has stood still inside the black hole. But, it does not mean that you will not feel any change of time. You will very well feel the passage of time.