Can I buy a new mac mini and then trade in my old one after I transfer to new one

Yes, if you arrange the trade while making an online purchase from the Apple Store, they send the trade-in box at about the same time as they ship your new Mini and you have 14 days from receipt of the trade box to ship it to them. More information in Apple Trade In - Apple

I don't think so. As I understand it, you need to arrange for your trade in when you purchase your new Mac. At that point, based on your description of your Mac, Apple will provide you with an estimate of value of the trade in. This will be confirmed on receipt later. They provide us with the necessary shipping containers and instructions. You will have sufficient time to do your Migration Assistant transfer before you have to return the old computer. Just follow their easy instructions and the process will go just fine.

Not possible unfortunately.

You will get more for the MacMini via private sale than any trade in offers.