Thank you for taking the time to register for this forum. I would encourage you to just make one post and put it in the appropriate bulletin board, posting the same request in multiple bulletin boards will not get you a bigger audience, nor will it expedite a potential response - in reality, it would most likely slow any kind of response.
Yes, you can create a contoured design. It would help if you let us know what version of VM you are using - Cut, LTR, PRO, DSR, or XPT.
If you are using PSV vinyl, you do not need to use a cutting mat, unless your intent it is not only contour cut, but to contour cut all the way through - which isn't really recommended. If you are using HTV vinyl, then you might want to use a carrier sheet, as there are times that the HTV will not properly grip the slick backing of the HTV in a consistent manner, causing slippage.
Loading a carrier sheet is not any different than loading vinyl without one. You still need to properly set your point of origin, and you need to ensure it's aligned to minimize drift. There is inherent slack using a carrier sheet, so you don't need to worry about that, unless it's going to end up hitting something as the machine moves it back and forth through the cut window.
It is entirely up to you whether you weed your design on or off the mat, it would be a personal preference, but if it's PSV, then there really is no need to use a carrier mat int he first place.
By your last bullet point, I'm assuming that you want to contour cut a single color of PSV after it's been weeded and taped. If that's the case, you can give it a try just be aware of a few things - a) you would need to have registration marks for the cutter to line everything up properly, b) your wide swath of app tape might adhere more than you'd like to your carrier sheet that already has some tack to it, c) your cutter wasn't necessarily designed to cut that level of thickness, therefore the amount of drag on the blade might be detrimental to your machine.
Most of us not bother to contour cut PSV, unless the design happens to be a square/rectangle, or triangle, or something else happens to be convenient as I'm using my straight edge and rotary cutter. Contour cutting is typically used when printing, then cutting.
I'm sure someone else will also have a suggestion or recommendation as well.