We removed our old counter tops ourselves. Ours were ceramic tile over a mud and chicken wire base on diagonally placed boards. It was quite the mess and quite the enterprise. There was no damage to the cabinets from the process.

The ideal sequence would be:
remove old counters
prep and paint cabinets
install new counter top

You could always put on a temporary plywood or some such counter top if needed.

We had laminate counters and while we waited for the granite after templating, we just the put laminate countertop back on. (had to do that 2x time since the 1st granite was cut wrong). Even hooked up the sink again w/ the laminate sitting on the cabinets.

So I would definitely remove the laminate first.

You might want to pick out your granite before you pick out your paint. That way you can have a better match. Even whites come in an astonishing array of choices.