Well, you can do that this way:
- Delete that two faces you have selected in the image
- Enter into vertex selection mode
- Select four vertices (two of each side, of course), for example, the front ones
- Press F to form a face between that four vertices
- Repeat with the four top ones, the back ones, and the bottom ones
There you have the plate.
As for apply some modifiers to only part of the mesh… you’ll have to separate the mesh into parts. As far as I know, the modifiers apply to whole objects. To separate, select the vertices forming the seat and press P to separate.