Mat is the Tie that binds.  Mat wed Tuon, Tuon wed Mat.  The Dragon's Peace bound Tuon to the land and the people.   Tuon came to trust Mat always speaks truth and only says what he means. and when he vouched for Rand and what he was doing.. It held weight in Tuom's thinking.  Rand Knelts not to the Empress. he knelt to Tuon who wasnt sitting on the crystal throne.  She was in her garden.  Her place of peace.  He made flowers bloom as he walked.  And completed the prophecy..

Look up the different levels of psychic that are about.  Mat was a Diviner/Seer.  He could sense future events, he also had awareness that allowed him to sense people around him and their intent.  to answer Mat's bowstaff fighting skill.  He was that good, the two rivers boys learned long bow and were scary accruate.. when you have Rands father and Mats father teaching..  you better be good at focus, and they taught all the boys of two river fighting skills as they still had random encounters with trollocs. Why do you think in a battle against thousands of trollocs, two rivers held.  They are taught skills by empire trained soldiers who retired young.  In the short story about ravens when they were young teens.  sword fighting and pole fighting. its mentioned in the story.  Its once he died the first time and gained memories of all battles that his skills improved to the point where it didnt matter how much of a sword master you were, you were not going to take down Mat.  Master of strategy.  Rand has full psychic abilities, Perrin had my favorite. Astral Travel, and teleport. He could shift in and out of this plane.  that is a very difficult skill and he did it naturally.  

by the end of the series all 3 could think of either of the other and know what they were doing, and could know which direction they were.  psychic telepathic connection.  Alot of people have it today.  

Also if you ever tried the Wheel of Time RPG game, or dungeons and dragons, it goes into more detail about their abilities.  I think Wizards of the Coasts continued the story after Robert Jordan died when starting 5th edition, as the names have changed but the races and stories are very similar.   Just look at the show Critical Role. if you watch from ep1 of campaign 2 and listen to the story closely you can spot Eye of the World being told.  Mercer is a story teller equal to Mr Jordan.