The 4th-level cleric spell freedom of movement [abjur] (Player's Handbook 233) is notoriously vague. (See this question.) The most liberal reading of the spell allows the subject to move normally through walls, while the most conservative reading views the spell's even in the first sentence of the spell's description as and therefore limiting the subject to exactly and only the effects the spell mentions by name. (My experience has been that most folks fall somewhere in between.)
That means you should ask your DM. However, I can explain how I'd rule on the two issues the question raises.
- Must the subject of a freedom of movement effect make a successful grapple check to exercise the grapple option move? Yes. This DM would rule that just because the subject of the freedom of movement effect can automatically succeed on grapple checks to escape a grapple, here the subject is not. Instead, the subject is trying to bring a struggling opponent with whom the subject is grappling to a different place of the battlefield. The freedom of movement effect is of no help in this task.
- If the subject of a freedom of movement effect successfully exercises the grapple option move action, can the subject move its full speed? No. In this DM's campaigns a freedom of movement effect includes neither the ability to carry excessive weight and still move and attack normally nor the ability to move at normal speed while wearing medium or heavy armor. Hauling around a struggling opponent, to this DM, falls into the same general category as those two ideas.
Again, the freedom of movement spell is enormously unclear on what it covers, and a player should expect every DM to have made different house rules to enable the spell's functionality. One DM's freedom of movement is highly unlikely to be exactly the same as another DM's freedom of movement.
Note: My house rules actually say, "When there’s a question whether or not freedom of movement affects an effect, if the DM has no opinion, those present at the table table will vote, and the freedom of movement effect will affect or not affect that effect thereafter." My original plan for my house rules had me include a list of every possible thing in the game that the spell freedom of movement could and couldn't affect, but the list still remained unfinished after three pages. I excluded that incomplete list and included that blurb instead. It just seemed a better tack to take.