In this video it is said that Buddha walked on water. Is that true and written down? And why did he do that?
It also says that Jesus might have been trained in the ways of Buddhism? Is there evidence to support this?
There have been books supporting the idea suggested in your question ("Jesus might have been trained in the ways of Buddhism?").
Christianity was a religion not generated from a teaching and converted to text. If you study Christianity's origins as an outsider from a historical perspective you will see that Kings,Clergy & some other influential figures modified the religion to suit the needs and even the birth of modern Christianity is a bit of a untold story.
As Buddhists we do not speak bad of other people's faiths so what needs to be said about the suggestion you make is that there hasn't been direct proof of any kind to support the idea. And even if Jesus was a Buddhist he would not go around offering food and walking on water because Lord Buddha forbid the acts of miracles unless it was an extremely necessary task.
Let me tell you a story for why i'm saying this.....
One day a Shepard boy saw a monk walking towards him while going to work. He saw that this is a good opportunity for him to gain some good karma and he stooped the monk and offered his lunch to the monk. After he was on his way back to work but he was bit by a snake only after taking few steps.
The monk saw this and thought to himself "This kid could die any moment,the best help i can give is to be more worthy of his offering". So he sat down and started meditating just few feet away from the child. He was able to reach nirvana Before the boy died. The boy died happily looking at the monk. He opened his eyes in the higher realm called "Deva realm" (Buddhist version of heaven). He returned to earth to tell the Arahat monk his story.
As you can see for a true Buddhist monk the real suffering is not hunger,slavery or poverty. It is "Samsara" and he only care to help a being's Karmic circle to end so he would be closer to Nirvana.
Walking on water....
Walking on water is a piece of cake compared to what Buddhist monks and Lord Buddha has done.
The best one of all miracles that only a Lord Buddha can do is called "Ama maha pratiharya" which is basically spraying both fire and water at the same time from the whole body.
Here are some very unique miracles Arahat monks have performed.
Optical illusions - Making moon or sun appear closer and show as if he or she is touching the surface.
Walking through mountains.
Making waves from solid objects like earth.
Sinking in to earth like oil and appearing from a desired location.
Transporting things trough space (Almost like teleportation)
Shrinking body size to fit through a keyhole of a door lock.
Lord Buddha's view on levitating upon water...
One day Lord Buddha and a group of monks were waiting next to a river until the boat arrives. A monk or a sage from another religion saw this and he immediately levitated off the ground and crossed the river. This was a mocking.
Lord Buddha asked "How much does it cost to reach across on this boat?". That is how much his practice is worth.
Lord Buddha said this because the ability to levitate can save him from the fee but that ability does not help him at all on his long journey of "Samsara".
Lord Buddha saw Miracles as a non important thing and never encouraged anyone to follow them. Because Lord Buddha's work is not to teach people how to do miracles, Lord Buddha knew that there is no need of Buddhism to give people super powers because those existed before Lord Buddha even though Buddhism unlocks some unique ones of its own.
One day in the early age of Buddhism Lord Buddha saw a misguided group of sages and went to make them understand the wrongful way of their understanding.
Lord Buddha saw that the leader of the group is full of ego and the best way to make them understand is to get the leader to a correct understanding. So lord Buddha decided to spend some time among them. One day Lord Buddha went to have a bath and thought "It would be easier if there was a branch to hold onto when getting in and out of water" then the King of Gods ("Sakra Devendra") came and bent a branch nearby. After having the bath Lord Buddha thought "It would be nice if there was a stone to wash my cloths on", and the King of Gods came and created a large stone.
Not only Lord Buddha was capable of miracles everyone was willing to assist Lord Buddha in any way possible knowing the power of Karma they could gain. That's why you see a "King of gods" acting like a personal assistant.
Truth about Jhanas and miracles...
You do not need Jesus to walk on water even you can. But as Buddhists we do not pursue such worldly goals. And there is no necessity of Buddhism to do miracles. Id you go to india you can finds many sages and monk who never knew the first thing about Buddhism and doing pretty amazing stuff. But they all do it to be praised or to gain a following. That is exactly the reason Lord Buddha forbid the showing off part of the practice.
Because soon as you feel special and more capable than the normal humans what you do is very simple. You go out there and say "Look at me vermin, i' the god".read the following to see why...
Here is a good example from lord Buddha's time....
One day a monk and his apprentice monk went to worship a temple and the teacher said "I would love to offer some flowers to this great stupa.".So the apprentice monk took off to the sky and disappeared, Soon he came back with a bucket full of flower from the "Himalayas forest". The flowers were offered and the elder monk gave advice to the younger "Be careful with your mind because if you don't you will be looked after by a one eyed woman".
One day the younger monk (Apprentice) was going somewhere through a jungle levitating in the air. Then the monk heard a nice song sung by a lovely voice. He lost his "Jhana" and slowly descended to earth. He has lost all of his powers because of the attachment he made up within seconds to that voice.
After few days he married the girl who sang and became a householder he was a monk no more. He had become a monk very young and had little knowledge about the life of a householder. After starting a fight with his wife he threw a knitting needle at his wife which ended up piercing her eye. He did not intend to do that but suddenly he realized the teacher's advice.
The is an event monks to this day teach to us and to apprentice monks to remind us the temporary nature of all things including powers.