Almost all the rare breeds, and crossbreeds !
Would not recommend an iron age (a wild boar cross) if you have kids and not advise Mangalitza for a first timer, otherwsie go for it.
All the rare breeds and crosses of them will be quite happy in your woods. Prick-eared breeds (eg Tamworth, Berkshire) are more inquisitive than lop eared breeds, but we have had children in with all our pigs, and providing you spend time with the weaners when you get them to tame them down, and teach the children how to greet and treat them, it's a great way to get your kids outdoors, learn about animals, and reconnect to where food comes from.
As for ruining the woodland, they key here is regular rotation, which means getting your pigs electric fence trained, and then dividing the wood into bits which you move the pigs between. Many use a radial system, with an ark and water at the centre, and then the pigs have access to bits of it, like a pie chart. This way parts of the wood can recover. Electric fencing works well, as don't need to chop down tress to create a straight fenceline, saving work and preserving the wood, just strim a 6 inch wide wiggly path round the trees, and use plastic stakes to follow this with the electric tape. BUT your pigs must be trained to it first. our book explains this and lots more in details (other books are available)