How do I properly respond to a 30 day notice of default of my proprietary lease in a COOP.

Your situation is not unique and it has often - especially in ocean front and near ocean buildings with balconies. First the issue is whether the corporation (a co-op is a corporation and you are an owner of stock, not of real estate) has the right to demand you remove your tile on the balcony. There can be many reasons for this and your short summary does not designate the reason. Typical reasons would be that the balcony was tiled without permission - or without proper installation techniques - or that the tile installation is causing deterioration of the balcony (typically through retained moisture, and in seaside situations, enhanced salt retention) and thus a danger to the structural integrity of the corporation's property. These are typically legitimate reasons for action by the corporation. The second determination is whether there is a reasonable necessity for the action. This relates back to the "why" discussed just above. The third determination would relate to whether your installation of tile is "grandfathered" in or whether there may be some time bar to the enforcement of the action. Included in this would be whether the action by the corporation is arbitrary as to some units and not others. A careful review of your co-op documents and the board action (through its minutes) to see how they got to make this demand upon you is necessary by an attorney so you can be informed before you jump into a lawsuit that could end up being unsuccessful and costly. You should not take any further actions until you consult with an attorney and you should do that right away as the "clock is ticking" on the board's timeline.