stove top covers are not stainless, they are chrome plated mild steel, hence the rust. Use naval jelly or evap-o-rust type products to remove it. Nothing rougher than a green scrubber pad is necessary to remove the rust if you use a product that dissolves rust.
Unfortunately, there is not much that will preserve the finish other than stripping and re-plating. However, naval jelly will leave a blackened surface (actually an oxide coating) that if not scrubbed too much will last quite well and prevent rusting.
BTW, the clips that hold the screws are called speed clips, and are available at McMaster-Carr.com, where they are called "clip-on nuts"