Is it legal to pick up cars from a previously claimed route and replay them elsewhere?
You cannot pick up cars from a previously claimed route in order to claim a new route. Resource management is one of of the control factors of the game; you only get 45 cars to achieve your routes. The game meets its ending condition when a player has 2 or fewer train cars remaining.

Review the rules for the game. They are written from a permissive stance, instead of a restrictive stance. For instance, the actions you can take on your turn:


  Draw Train Car Cards – The player may draw 2 Train Car cards. He may take any one of the face-up cards or he may draw the top card from the deck (this is a blind draw). If he draws a face up card, he immediately turns a replacement card face-up from the deck. He then draws his second card, either from the face up cards or from the top of the deck. (See Train Car Cards for special rules for Locomotive cards).  
  
  Claim a Route – The player may claim a route on the board by playing a set of Train Car cards that match the color and length of the route and
  then placing one of his colored trains on each space of this route. He then records his score by moving his Scoring Marker the appropriate number
  of spaces (see Route Scoring Table) along the Scoring Track on the board.  
  
  Draw Destination Tickets – The player draws 3 Destination Tickets from the top of the deck. He must keep at least one of them, but he may keep
  two or all three if he chooses. Any returned cards are placed on the bottom of the deck.  


The text is filled with "may" statement qualifiers. From that, if the rule book doesn't say you can take an action, you can't take that action.