I have heard of de-mucking happening right after draining, like John F mentions, and often is the only way IF the dam can not be cut deep enough to allow all the water to evacuate and you have to rely on a pump to empty entirely. My 1/4 acre pond had the dam cut in early June, all but a very small portion about 10 foot diameter (5 inches deep) drained. The de-mucking occurred later that year in early October and then had the consistency of very thick pudding. It would stand at 8 feet tall as they dug into it, but if let set overnight, it would slough off and ooze around. The summer was on the dryer side that year, but not as dry as this year. It was not until mid summer the following year before the muck on the back side of the dam could be worked with a skid steer and it still pumped a lot in a few place and had to set for a week or so to continue dressing the backside where it was extra thick/deep.
My pond was originally 10 foot deep and had silted in to only 2 foot deep. It was past due for renovation. How deep is your muck? If your pond was only originally a foot deep than it is now then you have to consider your options...de-muck out a foot and be done or consider digging into the original clay bottom to make it deeper. This option is risky because you are changing the "clay bowl" that currently holds water and the new clay bowl may not.
The best time to break a dam is after the spring rains, but any other time of year can work as well. After spring helps ensure that you'll get some dry weather to follow. So much depends on the weather and your pond excavator's schedule.
Time to allow for the muck to dry before entering with equipment depends on the skill of your operators and their available equipment, and the steepness of your original clay bottom. My pond had/has very steep sides and one miss step by the track loader and it would be at the very bottom of very deep muck in no time. My dirt man thought it best to wait several months.
Having spawning areas really depends on what type of fishery you want. A self sustaining pond will need areas for spawning, but a fish farm style pond where you feed and take and replace does not want shallows specifically for spawning. For example, if you want to concentrate on Hybrid Blue gill and Hybrid striped bass. The HSB will be used for keeping your low numbers of hatched HBG to a minimum and you do not want any small mouths to feed so that you can feed the mature gills and grow them without competition from younger fishes. This scenario does not want the promotion of the spawning of the HBG. The self sustaining pond will need some spawning areas so that the regular blue gills can reproduce so that the regular bass will have something to eat. The basic difference between the two is "are you going to feed hybrids or are you going to stock the pond and let it do what it will do?". Obviously, there is a lot more to it, but hopefully I have shed some light on the topic.