Our insinkerator batch disposal stopper can be placed to let water flow to the drain yet block all but small bits. It can also be set to stop water, but I find that it tends to leak unless I position the stopper just so. These two stopper positions are in addition to grind mode, which you twist either way to activate. When I want water to absolutely stay in the sink, I usually use a plastic tub or a inexpensive flat round rubber thing that you lay over the drain.

I have a batch insinkerator also it can be positioned to grind, strain or stop. I don't have any problems with leakage with mine.

I have a batch feed with a single sink. I usually leave the stopper out and I don't have any problems with things falling in. It happens occassionally, but it's not an issue. I turn my stopper upside down if I'm filling up the sink--it kind of turns into place.