Anchor - Yes, and/or stake out pole. Depending on the depth and current you plan to fish.
Mounting to the Kayak. Trust me, drilling holes in the deck, and mounting a rod holder will not harm the yak, and will be invaluable. I know, lost a travelling kayak rod and reel in Alaska last year because I didn't have one, and set the rod down for a moment... Long enough for a Coho to hit the lure and swim off with my expensive equipment.... Luckily I had access to a rod to borrow from the guide, else I would have ended that aspect of my fishing trip on the second day, and been forced to fish shallow with a fly rod the remainder of the trip...
Where to put the fish. Collapsible soft sided cooler. Take out the hard plastic insert, and it collapses until needed, and can be stowed in a variety of locations in the cock pit, or strapped to the deck. Have the Blue Ice cubes in it to keep the fish cold. Then, when you catch a fish you are going to keep, clean it immedaitely, or at least bleed it )By cutting the gills out) before putting it in the cooler on the blue ice. Then your fish will be super fresh.