Hi,
I don't know if there is anything out of the box for this solution, but depending on how your site is designed, you can probably create some custom tracking with your developer's support.
I am going to use Amazon as an example. Amazon has information about what is in a person's cart on every page (the user doesn't have to actually visit their cart to know what items, how many items, etc are in their cart).
In theory, if you added some s.product tracking to each page about what is in the cart (using a custom event to distinguish it from your cart views).
Now, your developers are tracking information about what is in each user's cart, so they should be able to add information about the date each item was added... unfortunately, Adobe dimensions cannot store numeric or date formatted variables... everything is stored as text. So you might want to add Merchandising eVars that store the added data as a standard format that can be cast to date in external program, but you may also want to have your developers calculate the number of days since the item was added and provide you that number... so that you can track "product a" as 5 and "product b" as "9", etc.
These values (5 or 9) would still be tracked as text, but visually you can show the Unique Visitors by this dimension (note that the same UV would be counted under each product length), you can create classification buckets to pull these as "less than week", "1-2 weeks", "2 weeks to a month", etc...
You could also do a similar "cart opened date" and "days since cart was created" as standard (non-merchandising) dimensions.
These merchandising and non-merchandising dimensions can be tracked on all pages with a custom event to use for your scenarios 1 and 2 (how many people have items in their cart, how long have those items been there), as well as being set on the purchase event to satisfy scenario 3 (how long were these in the cart before purchasing). The non-merchandising "cart" values would work for scenario 4.
