I know that NFL exclusive franchise players can't sign with other teams during the contract, but when the contract expires does their original team still have rights to the player? what are some other restrictions that exclusive franchise players face due to their contracts?

If a player is an exclusive franchise player, than they are unable to negotiate a contract with another team. The designation only lasts for one season, and after that, it is up to the team. They can deem them an exclusive rights franchise player again, which would prevent them from negotiating a contract again.

Walter Jones was franchised by the Seahawks three years in a row.

Exclusive just means for that year. The team could still trade that player away but a contract would have to be worked out first. Teams can also remove the tag ....only used if they want to sign them to a long term deal or trade them. After that year they are an unrestricted free agent. Look at Adam Vinatieri for example... after the Patriots tagged him he signed with the Colts.

Players under the Franchised tag can be signed by other teams but those teams have to give up 2 first round picks and the risk of injury is to high so the other teams tend not to give up 2 first rounders. the tag only lasts one season and players typically hate it because of the fact that if the get hurt during that season they get a minimal contract or nothing after that if it is a career ending injury.