There's something up with your terminology. There isn't even such a thing as integrated counter depth, though I take your meaning, so to make sure we're talking about the same things:

• Full depth fridges are free standing. The depth depends on things like how much air space you need behind and if you're counting the handle, but they're in the neighborhood of 33" deep.
• Counter depth fridges are free standing. The same caveats apply about the depth. Most are around 26-27" deep.
• Built-in are, um, built in. Attached to the walls and cabinets. Without handles they're usually 24-25" deep.
• Integrated are a special subset of built-in. They are attached, and their panels meld right into the cabinetry so that the shape, or styling differences (e.g., different color door or larger handle) is the only giveaway that it's a fridge. You can put divided cabinet panels on them to make them disappear entirely.
• Panel ready means accepts custom door panels and hardware. It is available in some models of all kinds of fridges, though most models of built-ins have a panel ready option.

For the antipasto tray, many people keep the old fridge in the garage.

You can make a full depth fridge look built in, or at least counter depth by recessing it into the wall. There may be a few that can even been made to look integrated, but there might be compromises to the door if you do. I'm not sure if you meant integrated, however, since counter depth aren't... Maybe you meant paneled?