What to look for when buying bicycles for children
To answer your specific questions first:


wrt weight: you are paying less for the children's bikes than for your own I hope? (they'll only fit on it for a few years after all) so kids' bikes use cheaper materials, cheaper usually means heavier in cycling gear. Don't expect to match your racer, but twice as much is a bit much even for a cheap model. Shop around a bit more.
Tires: Wider tires are more stable because of the wider surface and because the higher tire weight that comes with width increases gyroscopic stability of the tires. Width, and the thickness that comes with it means the tires can take more of a beating going over bumps, ridges, ledges and sidewalk edges. So actually wide/heavy tires save you time in money in patches both for your kids and for the tires.


As for things to look for:


Make sure the bike just barely fits your kid now (i.e buy maximum workable size), that way you can use the bike for a few years with just some adjustment to handlebar/saddle height.
Make sure you can add something like a flag on a long (high) pole to the bike. That way your kid is still visible from the other side of parked cars etc.
Brake quality is a big issue, especially if you are going out on the road with your kids, more so if the route is slightly hilly.
avoid bikes with gears unless you like spending your weekends cleaning and fixing bikes. Less gears=less maintenance.
Regarding the tire issue, and my point about bike size: when considering multiple bikes: get the bike with the largest tires. Larger tires=higher gyroscopic force=more stable and all at the same weight.
Don't overspend. Your kid is not going to be as careful with his/her bike as you are. You shouldn't be wincing every time they just let the bike fall to the ground where they stopped.