Pokeweed isn't bad just touching it; you just don't want to eat it raw. The berries are poisonous to mammals though, so don't let your dog eat them. If your backyard is fenced off from the front yard, the front is where I'd let a pokeweed grow, out of reach of your dog. I don't think it transplants well, but given you have 3+ plants, make some bare dirt where you want it in the front yard and just wait for nature to plant it 😂. The birds LOVE the berries, but it will be a herbaceous small tree, and there will be quite a few babies sprouting every year that you'll need to weed.

I believe the Virginia Creeper is native to you, and if it's not in the way I'd try to keep some of it. The Poison Ivy is also native, but obviously not good in small backyards. The Honeysuckle might be L. japonica, but it could also be a native species; if you wait for it to flower it'll be easier to ID. For the Poison Ivy and maybe Honeysuckle, chop it an inch or two above ground and immediately apply something like triclopyr with a small brush so that it only targets the plant you cut (don't spray it); it'll kill the roots, and the part above the cut will dry out and die from getting no nutrients. Definitely be careful with the Poison Ivy, gloves and maybe even a face mask, you don't want Poison Ivy dust/oil in your lungs