Rumford, seventh at last week's Ballantine's Championship, holed his approach at the fifth for eagle but double bogeyed the next.
But three birdies either side propelled him to the top of leaderboard after eight holes at El Prat.
Wall birdied the second, fourth and sixth to be three under through seven holes, one ahead of a group of nine players.
France's Romain Wattel and Christian Cévaër, Ireland's Peter Lawrie, South African Thomas Aiken, England's David Horsey and home trio Ignacio Garrido, Jesus Maria Arruti and Pablo Larrazábal were all two under early in their first round.
Miguel Angel Jiménez, Colin Montgomerie and Jose Maria Olazábal were among the later starters.
France's Wattel was only one under par after 11 holes, but birdied the 12th, 15th, 16th and 18th to complete a flawless 67 and set the early clubhouse target.
Spain's Jesus Maria Arruti was also five under with five to play, while France's Gregory Bourdy and Ireland's Lawrie were in the clubhouse on four under after rounds of 68.