board approval for renovations???

Depends on your coops by-laws but I would imagine that they will only need to insure that your contractor is licensed and insured.

It is a case by case basis.

In most co-ops (being a past co-op board president), if the work you are doing is completely nonstructural (and thus deemed design work), the board will STILL have to see all fully executed plans first from your contractor/deisgner/architect as a sheer formality to assure themselves that what you are doing requires no more of the board's attention.

In the case of structural work of any kind, you will most certainly require full board review and approval. But if all you are really doing is changing cabinets and drawers, and not moving the gas or water lines or appliances, I would simply ask a board member in your building if that even demands a review or not.

we're buying a top floor coop apartment and want to install central air. What if any tips do folks have about getting board agreement on placing the compressor on the roof (above our flat)?

This will vary greatly, depending on the board. If you have NOT purchased the coop yet AND the central air issue is critical to you, I STRONGLY suugest that you have a discussion with a board member or two in ADVANCE of your purchase. Do NOT rely on what the broker or current owner tells you regarding how the board will react. The board will have to consider the issue of breaking the roof membrane, the noise issues generated by the compressor and if this will inconveience any other penthouse owners (especially if they have outdoor roof terraces), etc.

Thanks. There are no penthousea.. and my guess is that there's a way to place the compressor on the roof without breaking the membrane (the pipe to the air handler can be routed up to the roof on an outside building wall--in the back of the building). What, if any, proprietary issues could come up?
Thanks

I read in New York Magazine, I think, about a couple who purchased 2 adjoining studios to join for their new home. After closing it was reiterated that yes, they could join the two apartments - but would have to put them back the way they were when they sell! I don't know if that's even a true story but I'd want to get everything in writing before I close!

most likely not true! if you are joining two studio apartments to make it one apt, you will need both the board and the office of building's permission to do it. The Office of Building controls any conversions. Asking for a permit requires extensive paperwork, inspection, etc. That would take months... And how in the world are they going to bring it back to its original fixture without having to go through the office of building to file additional paperwork to convert it back to two studios???