The Zanarkand that Tidus is from was a dream. If the dream comes in to contact with Sin then there is a chance that the part of the dream that came in to contact with Sin can become something more than just a dream.

However Tidus and Jehct before him were still only dreams but instead of existing in the flow of the dream they existed outside of the flow in the world of Spira (reality)

This would be akin to you dreaming up your fantasy of the perfect girl and the next day you awake and find her sitting in your room completely real so that everyone can see her, touch her and speak to her but then one day you stop dreaming about her and she disappears.

That is what happened to Tidus and Jehct thanks to Sin. Jecht however complicates it by becoming the fayth for Sin a way that proves that whilst he is in Spira he is as real and more importantly from the way this conversation is going, alive. Tidus follows suit apart from he doesn't want to become Sin (for that was his other choice I think he would have had to have been the fayth for the final summoning for Yuna had that been their choice) The dead however can freely ride Sin to the dream and back or rather they can if Sin allows them to. That is what Auron did to get to Tidus and to make sure that when Jecht (Sin) was ready he would take him back and guide him to his destiny.

It was when the dream ended that Tidus's existence ended when you awake from a dream it is as if the dream never was, but as I said about the fantasy girl, those she spoke to and touched would forever bear the witness to the fact that she was there and real. Hence Yuna and her other guardians (not to mention half of spira from the looks of it in the game) will remember Tidus as being there even though he disappeared along with Sin and the Fayth.

Now as for Tidus's looks and existing 1000 years before the game's events the people here are correct Tidus didn't exist 1000 years ago in Zanarkand. In fact Tidus's appearance and such I believe was based on someone from the real Zanarkand however (as you will see if you play X-2) called Shuyin who like Tidus falls in love with a Summoner and will do anything to protect her and prevent her from dying.