It is probably like that to prevent losing your hand, such as Mace Windu, Anakin and Luke Skywalker have..

Remember when Kylo used them to burn Finn? I felt like that was added in just to stick it to everyone who thought they were stupid.

I hear a lot of people complaining about this, but there are totally valid reasons as to why he lost the fight.

1. He had just killed his father. He was clearly shaken by the action. This is confirmed in the novelization, where he is described as "feeling weakened."
2. He had suffered a bowcaster shot. This is established twice in the movie as being an incredibly powerful weapon that blows people off of their feet. The fact that Kylo was still able to fight with a bowcaster wound proves that he's actually extremely tough.
3. Kylo had not yet finished his training. Not to mention, for a long time, Kylo is the only guy with a lightsaber running around the entire galaxy. The New Jedi Order had been dead for a while and Luke Skywalker had vanished. He hasn't exactly had a lot of dueling practice as of late.
4. He was toying with Finn in the beginning. You can tell because he gives Finn time to recover and does some lightsaber flourishes to show off. It wasn't until Finn managed to get a hit in that Kylo immediately dispatched him in a few hits.
5. When he's fighting Rey, it's important to remember that Snoke had ordered him to bring her to him. So he wasn't going into that fight to kill her. He was also caught off guard at this time because he wasn't prepared for Rey to become as in-tune with the Force as she did in the last minutes of the fight.

There. I'm really surprised at how many people think Kylo was snuffed; there are plenty of really clear and understandable reasons why Kylo lost that fight. He's an incomplete villain. That's what makes him interesting. We're going to see this guy grow in the next two movies and I guarantee you he's going to have a lightsaber fight that he'll dominate in.

I don't know if I saw it wrong, but why was Rei's lightsaber blue? Wasn't Luke's last lightsaber green after the blue one got destroyed?

Maz Kanata somehow managed to get a hold of Luke's first blue lightsaber after it had fallen down the shafts on Cloud City. I expect we'll hear the story behind that in the next movie. I assume that Luke still has his green one.