Hi Shah,
Doug said most of what probably answers your questions. The TarpTents are great tents, but be careful up on ridges in the Alps. Like you've probably heard the weather there is notorious for suddenly changing and powerful winds and heavy rains are very common. It is usual for rains to come evey afternoon, and often snow at higher elevations (about 2,000 meters and up), even in mid-August (it snowed three times while I walked last August, once down to about 1800 meters). TarpTents are not designed for alpine storms, so you may want to give your wife and daughter a chance to get more used to being outdoors before they face an above-treeline storm. Also, be very careful of thunderstorms on the ridges. They are very frequent in the Alps (much more frequent than here in the Japan Alps). If lightning is in the forecast I strongly recommend getting down off the ridges and not camping up there.
From the way the tarptent is pitched in the photo it looks like you didn't pull the front corners out far enough. What I usually do it stake out the rear first into the wind, then pull the front apex out with the hiking pole set into the grommet, staking that out with the guyline, and finally pulling out the front corners and staking them down. Usually if the weather is fair lifting the front end of the tarptent up to allow airflow will work much better in airing out the tent, but the flat apex strut of the tarptents make for a fairly unstable unit in the wind, so then you may want to lower the front and pitch the whole thing right down to the ground. As Doug said staking out the long sides of the tarp make a very big difference. Just be careful not to pull them out too far or the roof of the tent begins to lower.
I think TarpTents are fine for the TMB, but only for the lower level campsites, which are very common along the trail. It's one of the things I really worried about all summer before I left. After having experienced the trail and gone through some bad weather (there were some torrential rains for four days when I first arrived that took out an entire mountainside just north of Courmayeur, with temperatures at 5º C down in Chamonix. I don't know what the trail was like because I didn't walk for those four days) I would now be comfortable with a TarpTent, but would make very sure that I was out of the wind. If you want to camp higher up I would recommend a more wind-stable tent, at least until all of you have more experience. Alpine ridges are no place to begin learning how to pitch your tarp. Practice a lot before you go. I've gone so far as to practice pitching my tarps outside my apartment at night during typhoons just to learn what to do in unusual… and extreme… conditions. It's one thing to pitch a tarp on a calm summer evening, quite another when the fabric is flapping all about your face and you have no idea where that stake went and your fingers are cold and wet and stiff and you are exhausted from a long day's climb!
One thing I think is important here on BPL is that more emphasis be placed on learning techniques and practicing them until you have them down pat, than so much on gear. The gear is important, of course, but it does no good if you don't know about good sites and weather and what to do in bad conditions.
Keep in mind, those ibex at the lower elevation campsite that you found knew very well what they were doing. There is a reason why they frequent that place and not higher up.