Did Joshua go up Mount Sinai with Moses?

Yes, Joshua accompanied Moses when he went up Mount Sinai, as the context indicates Moses and Joshua were together on the mountain. However, Joshua evidently stayed back when Moses entered the very presence of God, since Moses was alone with God. They then resumed companionship as they came down.

Les explains that when Moses went up into the mountain, Joshua accompanied him. But when Moses went into the very presence of God, Les feels Joshua had to stay back, because Moses was absolutely alone with his God. After God gave the instructions for the tabernacle and the law, Moses and Joshua picked up their companionship as they came down out of Mount Sinai. Later, Joshua heard the noise of the people shouting and thought it was war, but Moses clarified it was not a victorious army or people being overcome, but the noise of those who sing.

Exodus 32:17

And when Joshua heard the noise of the people as they shouted, he said unto Moses, There is a noise of war in the camp.

When did Moses and Joshua come back together after Moses met with God?

Moses and Joshua resumed their companionship as they came down out of Mount Sinai. Joshua had accompanied Moses up the mountain, but he stayed back when Moses entered God’s presence alone. They came back together on the descent from the mount.