Do we need to believe in Buddhist stories to be a real Buddhist?

Faith is for fools and advanced practitioners. That sort of practice is called "guru yoga" in Tibetan Buddhism. There they require you be an advanced practitioner before initiating you into that practice. And, in the past for the monastics, the guru then was responsible to provide food, shelter, and teachings for the person taking the vow. There is nothing wrong with being a "fool" for a particular religion (or more probably a particular teacher) but,if one is going to do that, one had better make sure that one is placing one's faith in the right teacher. I don't find "blind faith" useful in my practice. If Buddhism has truth (as I think it does), I don't think it needs its practitioners to practice blind faith. If blind faith is necessary, I will probably wander off somewhere else.

Trust is built on verifiable statements or personal experience. The "Western" students seem to respond better to examples of holiness, rather than descriptions of holiness. (If this method really works, then there ought to be examples existing today. And there are. Just really hard to find, for some reason.) Actual meditation practice, like fitness, seems to follow the "you get out of it what you put into it" policy. "Personal Experience" is only applicable to the one having the experience, but is no less useful. Ultimately, it's all about what works for you. Buddhism has quite a few traditions (and sub-schools). One of them may be for you.

A real Buddhist is a person who takes refuge in the Buddha, Dhamma and Sangha. Dhammapada, along with it's stories are a part of the Dhamma. So if you reject the stories and still claim to be a Buddhist, you are essentially saying that you don't accept those stories as a part of the Dhamma. But it's a bad practice for a person to start deciding what parts of the Tipitaka or commentaries belong to the Dhamma. Especially someone who's neither enlightened nor omniscient. Because that habit can leak into much more important teachings as well. Like questioning if there's rebirth or Nibbana. Because you don't yet know them by direct knowledge, just like psychic powers. So you are taking them on faith.

There's a way to attain psychic powers and it's explained in the text. Just like the way to attain Nibbana. So why not take psychic powers on faith too? It's not like you can disprove them.

Vicicicca(doubt) is one of five hindrances obstructing the path. You are rejecting the stories because of your Vicicicca. Not because of your direct understanding that the stories are false. So you are actually empowering a mental state which is harmful to the path instead of gaining any real knowledge about the authenticity of the stories. On the other hand, if you can simply accept them on faith, Vicicicca has little room to enter your mind. That'll also make you less distracted when you listen to the Dhammapada.

You should not believe all the stories to become a Buddhist dear friend. Instead you should believe Noble Eightfold Path and taking refuge in the triple Gems (Load Buddha, Dhamma and Sangha). Taking refuge doesn't mean that the person has to pray for them or looking forward until they do something for you. We have to believe Lord Buddha, Dhamma and Sangha then we have to follow the path which is showed by them. They are guides and helpers so followers have to follow the path.

If someone loose their courage to follow the path due to the stories that is the best enmity one can do for him/her self. First clarify everything about Noble Eightfold Path and follow it. Then when you gaining wisdom due to the follower, you will be able to identify whether the stories are true or false.

when talking about teleportation, we are in a era of quantum teleportation. even now with the existing technology we can perform teleportation on quantum particles like photons. but the problem is on the communication channels. if we divide our body into quantum particles and begin to teleport them, it will take millions of years to complete. If scientists develop a mechanism to build high speed communication channel with using quantum entanglement, we will have a teleportation era in near future. So don't categorize stories just as fairy tails or believe at once. Remember stories and follow the path then you will find facts do decide.