I've heard some people answer this question in the negative by saying that laughter is always a sign of frivolity and a thinly veiled attempt to make light of things that are sober. They say life is a sober matter; Jesus is described as a man of sorrows. He's described as one who was acquainted with grief. He walked around with enormous burdens upon him. Add to that the fact that there's not a single text in the New Testament that explicitly says Jesus laughed. There are texts, of course, that tell us he cried. For example, John 13 tells us that in the upper room Jesus was deeply troubled in his spirit. We know that he experienced those emotions, and yet it's strange that nowhere does it tell us that he actually laughed.
You also asked if he had a sense of humor. When we translate any language into another, we will often miss subtle nuances of speech. If we don't have a knowledge of the original language and its idioms, we might miss the humor. Also, different cultures have different ways of being humorous. Jesus used one form of humor we call sarcasm. In his responses to Herod, for example, he called him a fox and made other statements that I think had a touch of oriental humor to them. It's purely speculative whether or not Jesus laughed, but I can't imagine that he didn't laugh for this reason: He was fully human, and he was perfect. We certainly wouldn't attribute to Jesus any sinful emotions or forms of behavior, and it would seem to me the only reason to think he didn't laugh would be if we first came to the conclusion that laughter is evil.
The Bible does say that God laughs. In the Psalms it's a derisive laugh. When the kings of the world set themselves against God and take counsel against God, it says that he who sits in the heavens shall laugh. God will hold them in derision. It's sort of a "huh!" kind of laughter. It's not a jovial response of happiness, but nevertheless it's laughter.
In the Wisdom Literature of the Old Testament—for example, in Ecclesiastes—we're told that certain things are appropriate at certain times. There's a time to plant, a time to reap, a time to build, a time to tear down; there's a time to dance, a time to sing, a time to laugh, a time to cry. Since God has, in his seasons, appointed appropriate times for laughter, and Jesus always did what was appropriate, it would seem to me that when it was time to laugh, he laughed.