Yes, this is pokeweed. You don't have to worry about the toxicity unless the neighbor's kids are in the habit of eating weeds. Contacting it is not a probem.

Yes, it probably has a huge tuber underground storing a lot of energy, which will continue to sprout foliage. You can simply cut it down repeatedly, every time it attempts to put up a shoot, until all the energy stored in that tuber is exhausted.  That could take a long time, but it looks like it is in a lawn area that would get weekly mowing anyway.  That should do it. The aim is to not let the leaves rejuvenate the tuber. 

Another option (or combine with the cutting down) is to apply a systemic herbicide that will go down into the roots and kill the tuber.  The ingredient glyphosate is what you would look for on an herbicide label. If you are careful to spray only the pokeweed, you would not need much. 

Glyphosate is a total vegetation killer, so do not let it contact any desirable plants, like the grass nearby. Use a shield to keep spray off other plants, or paint it on the pokeweed's leaves. 

Here's more about pokeweed: https://extension.umd.edu/resource/pokeweed

Ellen

We have not encountered problems, but can't guarantee it either as there is vast variation among humans in their reactions to everything. At any rate, since you are removing it, there should be no problem. 

Our pleasure.