Can every martial art have a sport aspect?

Sure. But usually the sport aspect can only be a mini-game based on rules derived from a very limited part of the said martial art. Like the mini-games you often use during the warm-up of a training session. You can't have a sport which exactly represents all the content of a martial art, since by definition a martial art is originally created for war, not for training with friends you don't want to hurt.

For instance Judo is one of the sport aspects of the martial art Jujitsu, together with Brazilian jiu-jitsu and "jujitsu fighting", which are other sport aspects of the same martial art.

No. Most do, but not all. For example, krav maga techniques are way too dangerous to be used in sparring.

Absolutely. Make rules that you can not seriously hurt each other and a fight has to go for so long, take out the lethal techniques and you got yourself the sport version of the style.

Most martial arts have a sport element because the desire to "play fight" without killing each other. That does not mean the true art is manifested in the spot arena. Though, in real karate, all techniques are made for self-defense so you see people get injured. Look up some YouTube video of karate kumite or even Shotokan karate kumite.