Echoing others on here, I would definitely not go the insurance route and instead try and sell it on Craigslist As-Is. You could probably get at least $500-$750 for it, especially if it can be driven away. Alternatively, you could scrap it by either selling it or donating it - a scrap yard would love to have those parts in its inventory and they'll come and haul it away for free.
As for the replacement - you really have two options. Option one - you get another used minivan. Growing up in a family of 5 (3 boys, my parents), I can say that the minivan does offer the most utility when it comes to hauling people and stuff. Option two is to shop around for cars with a higher roofline and a deep backseat - a (used) Mazda5, Prius, PriusV, most Subaru wagone, etc. With a minivan, you have all the versatility and space you need, but with the smaller car you are getting more "mustachian".