You can configure a pop-up to display specific deal properties for completion when a deal enters a specific stage - however, I don't think this what you're looking for as it can't be applied to properties on the contact object.
What exactly do you mean by display? You can have properties that you show/hide at will in lists and such. You can have a property that just sits there unless someone looks for it or to use in AND conditions.
What you're describing seems to be what HubSpot uses Deals for. Is there a reason they wouldn't work? Leads are like timelines of actual people's activity so that's where you could look at the complete timeline including deals and what the stage/outcome is/was.
Also knowing what suite(s)/tier you are on would help know what tools you have access to.
You can configure a pop-up to display specific deal properties for completion when a deal enters a specific stage - however, I don't think this what you're looking for as it can't be applied to properties on the contact object.