When our kids were younger we always stayed in JH- in fact still do when we visit.
I should probably create a top question under Jackson for this, because it gets asked a lot.
The Town of Jackson definately has its perks in the summertime because it is a tourist mecca. There are a lot of dining, shopping and entertainment options to take advantage of after a day of hiking, biking or rafting. There is more to do than if you would stay in Teton Village, but some people dislike the busyness of it because it takes away from the peaceful park experience that just had. Just a thought. For kids, there is the rodeo, Snow King Activities (mini-golf, horses, alpine slide and soon to be zip line). There are bike paths in the town of Jackson, going to Teton Village and also, going to GTNP and in the park itself.
Teton Village is a little slower, since they are geared towards the winter season for Jackson Hole Mountain resort,, although all restaurants will be open as well as their shops. I prefer the dining at Teton Village over Jackson, but the shopping in Jackson is a bit more diverse. Teton Village has really stepped up their summer activities lately and enjoy taking my child there. Without going into detail, here is the link for more info http://www.jacksonhole.com/summer.html.
All of the activities you mention can be accomplished regardless of where you stay. If you have a car, you'll inevitably be visiting both locations so perhaps create a base around your housing preferences first. I hope that helps get you started.
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