You can see the Clipboard size for requestors in SMA (Requestor Management page > select Requestor Name > click the Clipboard Size button). You can also see the Clipboard size using the Performance tool. I am not sure about modifying the size.
As far as I know, the only way to alter the clipboard size is to add/remove data from the object in the clipboard. I'm assuming you're more interested in keeping it a smaller footprint. I'd start by looking for duplicated objects. If you have a page Foo with the pzInsKey of "YourClass W-1" and a page Bar with an object that has the exact same pzInsKey, you want to get rid of one of them. This keeps things cleaner, the footprint smaller, it also helps avoid potential data integrity issues. The second way to reduce the clipboard size is to remove pages when you don't need them anymore. If you needed the contents of a page to make a decision for a piece of work, once the decision is made, if you don't need that data anymore, remove the page. Same goes for the contents of pxResults when you run a report or lookup list. If you don't need the data anymore, remove it from the clipboard.
The good/bad news is that it really comes down to application design. You have the power to design a lean application, but it's incumbent on you to know when you don't need data on the clipboard anymore and to clean up after yourself. If you don't, your clipboard can quickly grow messy.
>>> If you have a page Foo with the pzInsKey of "YourClass W-1" and a page Bar with an object that has >>> the exact same pzInsKey, you want to get rid of one of them. This keeps things cleaner
You're in trouble if you ever have more than one page with the same pzInsKey. There are places in the core Pega level where it uses the "findPageByHandle" to find the page, and if you have more than one, it will sometimes pick up the wrong one and give you inconsistent results, and you can end up with stale data.