If the universe is infinite, then time is also infinite. If this is true, then why isn't the whole sky always light because light from the furthest stars have had an infinite time to get to us?  And space is infinite, therefore there must be infinite stars in every possible direction, meaning that the luminosity would have to be pretty freaking bright.

Think of a giant warehouse, say the size of two football fields. If we are in one corner and we light a candle it will give us enough light for out little area. Now if someone lights another candle the distance of one football field away we can see the light, but it isn't bright enough to light our space. Then if someone lights another candle all the way at the other end, again we may see the flicker, but it does not light our space.