Can I regrade this gravel section of yard myself?
Given the willingness to devote time and energy to the project, sure you can regrade it, but looking at the photographs it may not solve the root problem.

It appears that water will sheet flow down the driveway and directly into the building...that's why the drain is there. The presence of the french drain suggests that the problem predates the gravel extension.

Diverting the sheet flow well uphill from the building  will reduce the amount of water hitting the building. This means regrading a portion of the driveway (perhaps prior to the asphalt to concrete driveway transition would be easier).