The above is a dangerous setup. I'm wondering if they did that just to take a picture. Having only coaster brakes would not be sufficient to stop the momentum built up by that much weight in motion--at least not in any reasonable amount of time. Also, it must be a nightmare to get that moving with only a high gear to push.. but I think you can tell that from the guy's expression! I'd also be a little worried how that child on the end is hooked up to the tag-a-long with the sideways-reaching arm.... seems it may end up far out of line on a wide turn.
As to the OP - I've done the tandem + tag-a-long. My stoker weighs around 100lb and the child weighed around 50lb. Riding it felt like just having a heavier stoker than normal (as if the weight of the stoker was simply increased by the child + tagalong weight). Turning radius was a bit wide and unwieldy as well, even when compared to a normal tandem turning radius. However, turning wasn't a big deal as long as you plan ahead. We all got a kick out of riding it because it was so comically long and awkward.
As for adding a trailer in addition to that.. well..
I've seen the tandem+tag-a-long+trailer setup on a club century. It just doesn't seem like a great idea to me. You have two pivot points and an extremely long turning radius which will make it very unwieldy to control. The baby trailer in particular makes it difficult because the two wheel design makes it more as if you are driving an extremely long tricycle in terms of handling. I think you should consider other possible configurations that may result in a more comfortable, safer, and thus more fun ride, e.g.,
1. tandem + 2 child tagalong (like a tandem tagalong)
this would eliminate one of your pivot points and keep it to inline wheels.. however the youngest might have to grow a bit to fit this
2. tandem w/ kid stoker kit & 2nd bike w/ trailer
two bikes with similar weight and handling. shouldn't be hard to stick together. As time passes you can take off the stoker kit (so the older can ride on the tandem still) and swap the trailer for a tagalong as the kids grow.
3. 1st bike w/ tag-a-long & 2nd bike w/ trailer (no tandem)
You could do this until the youngest can ride the tagalong, then buy a tandem. disadvantage is that a bike + tagalong doesn't handle anywhere near as nice as a tandem for two people. Advantage is that you have more configuration flexibility and the setup is more easily transportable.
I'm sure you can think up more than the above. Just anything that gets you away from a bike setup that is 15ft long with multiple pivots and two wheels thrown in at the end. IMO it's better to just have to bikes that are reasonable to handle.
One last thing.. if this is just for <5 mi down to the park (i.e. brief transportation) I think the long trailer setup is fine. I just don't think it would handle well enough to be fun to ride for any length beyond that. If it is very rare that you actually take all the kids at once, it may also be an ok option.
edit:
here are some interesting pictures for large family tandem-style setups:
http://www.precisiontandems.com/artkidbackinstall.htm