What is the ruling on one who misses some rak'ahs of the prayer under the pretext that its time has not yet fully begun, even though there is ample time to reach the mosque? And is this considered negligence in prayer?

The first and best approach is to hasten to worship and to attain the first row and the opening takbir. This embodies the saying of Allah the Almighty: "So race to all that is good." One should encourage and urge a friend to do so, knowing that he is not sinful as long as he catches the congregational prayer, even if his action is contrary to what is superior.