I am assuming you have bosch dishwasher. Bosch came with this clever idea to make the washer drain hose with 3/4'' and most air gaps inlet are 5/8'' which is obviously large to clamp. Customers are forced to buy unnecissory kit which in the first place should come in the package. The only luck you have is to use the high loop drain system and will not have those beautiful air gap holes anymore. Otherwise you can just buy a drain hose with 5/8'' end to connect to your air gap inlet and you have to dismantle the back of the dishwasher and connect.
I did it myself quickly and easily. Not sure how much installation would cost, but it takes about 5 minutes max to hook up this connector so I can't imagine much at all.
You can cut to fit!
I believe all garbage disposals are the same same size where the drain line from the dishwasher connects, so I don't think you'll need this.
The supported sizes are 1/2, 5/8, and 3/4 inch dishwasher line input line sizes. Notably, the 7/8 inch size is NOT supported.
To my knowledge yes it will.
How to you remove the plug for connection to the dishwasher
This does not contain a check valve. If you require a check valve, you must add it separately. This is just a "step-wise" funnel shapped chunk of rubber, intended as a nearly universal adapter between various dishwasher output hoses and the insinkerator garbage disposal input barb. I say "nearly" universal because it supports the 1/2, 5/8, and 3/4 inch output line sizes, but does not support the 7/8 inch dishwasher output line size (which was once common, but is less so now.)