In a perfect world, you'd want to stick with in-phase or out-of-phase until you get a feel for riding a tandem. It would take a lot discipline on the part of you and your stoker(s) to stay that way but it would make the handling more predictable during your initial outings since each type of phasing creates a different "feel" to how the tandem handles.

However, that's not what you bought into so you'll just need to stick with it and wait for you and your stokers to come up the learning curve on how to pedal smoothly. In this regard, try to do most of your riding in gear combinations that are easy to spin at a relatively high RPM, e.g., 90ish. Keeping the revs up tends to eliminate pedal mashing and the associated side-to-side body English which I suspect may be part of what you're experiencing... but it's just a guess. The "death roll" that your brother mentioned was, as Zona intimated, exacerbated by his movements on the back of the tandem. Similar to riding two-up on a motorcycle, what you don't want are passengers who tend to be back-seat drivers as they can really muck up the works... particularly if they are as tall, taller and/or heavier than the captain.

Starting with the younger and smaller kids on board would be accomplished just as you did with your 14 year old: let them saddle-up like they would a horse and have them give it their best effort in terms of providing power as you start. Up front, just ignore the presence of the stoker and start out just as you would from a dead stop on an incline (exept that you'll want to keep the tandem perpendicular to the ground vs. leaning it over as you would your single bike)... use a short gear and get your down foot up and on the pedal as quickly as you can. Don't worry about "clicking in" until the tandem is carring enough momentum to coast in a straight line and then readjust your foot to engage the cleat.

Unless your spouse is similar in size to your brother (height more so than mass), stick with the 14 year old as your stoker using the higher cadence and practice coordinating your ability to stay locked in the same phasing... this is a still you'll want to have when you introduce your spouse to the tandem. Aside from that, easy as she goes. Be an attentive listener and encourage real-time feedback from your stokers on when the tandem feels unsteady so that you can try and correlate it to what you're gearing and cadence is and/or what type of riding situation you're in, e.g., climbing, slow-riding in congested areas, or losing focus while fiddling around with something else or sight-seeing.

It's just a big bike: just do whatever you can do to relax and to put your stoker at ease.... which does put a high demand on patience.
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