Suspension in kiddie Balance Bike

Honestly, it's largely a marketing gimmick putting it in. Balance bikes don't need suspension, let alone rear suspension - you sometimes see kids using them in the practice areas at trail centres and even they don't need suspension (though they sometimes have foot pegs).

The rear suspension looks like just a highly simplified MTB suspension, bearing in mind that there's no front triangle or space forward of the saddle for the spring and damper, but there's also no need to take into account things like chain length shifts as the suspension moves, or the rider standing on the (non-existent) pedals with their weight well back. MTBs don't use suspension like this because it pushes the rear wheel further back.

As for the front, I'm not sure what we're seeing there - maybe a flexible rubber stem (something sitting on it doesn't help with clarity). Stem suspension is a thing (reviews at CyclingAbout.com), and Specialized also have a suspension steerer tube. These are mainly meant for things like gravel bikes, for taking out the many small bumps that over time stress your upper body. They don't do much for big bumps, which is why mountain bikes have much more travel using suspension forks, and the one pictured would do even less. That's also at odds with rear suspension - after all a lot of mountain bikes are hardtails and go over far rougher stuff than this.

You mention seat suspension, and of course suspension seatposts do exist, as a comfort thing. My daughter has one on the back of the tandem; they're popular there because bumps you can't anticipate exactly tend to be more uncomfortable. These have short travel. On a bike where you can't take your weight on your feet over bumps, they may provide some benefit, but balance bikes tend to be used for short periods and kids, being lighter than adults, don't bounce so hard on the (anyway rather squishy here) saddle.

So no real upsides, and anything like that adds weight. Weight, for kids bikes, is a big issue. While a really heavy bike might get to 25% of my body weight as a tall adult, some of these gimmicky kids bikes are over 50% of the bodyweight of their target market, which makes them hard to handle and hard to pick up off the ground.

Rear suspension will also make learning to ride a balance bike harder, as the back will bob up and down as weight transfers from the feet on the ground to sitting on the saddle. If the biek is even slightly leaning, this makes the centre of gravity move. The only way to avoid that would be to have the suspension so stiff it's not suspension.

Learning* on a balance bike requires a surface that the rider can always reach with a foot, so for little kids a maximum of something like 5cm (2") change in surface height between the feet and either wheel. The limit you'll reach sooner is less clear, and that's the size/steepness of bump that the rider can (confidently) roll up. Bigger wheels help with that but not enough. Adding suspension doesn't really help on a balance bike (unlike one with pedals), and adding weight makes it worse.

Overall, there are no conceivable conditions in which suspension will help a kid learn on a balance bike, quite the opposite. And that's why none of the established brands do it.

* A confident balance bike rider may enjoy rougher stuff later, but that's another matter.

I'd like to address the three questions separately.

1. I read some articles that rear suspension in bicycle is for maneuverability and not more on rider comfort.

   Well this is sometimes said, but it's quite obviously wrong. Of course suspension is good for comfort. IMO this is kind of a "we're big strong mountainbike men, we don't need comfort" nonsense.
   Only indirectly is there some truth in it: suspension allows you to go faster downhill, and that speed then makes the ride rougher again. But then you could as well argue that improving aerodynamics doesn't save energy!

   So yes, it can absolutely make sense to add suspension purely for comfort. There are plenty of reputable city / touring bikes that do this. One just needs to be clear that it's a tradeoff with price and weight.

2. In kids balance bike, the kid's feet are directly on the floor due to lack of pedals instead of bending the knees when using pedals. So the vibration from bumpy floor can reach the kid more. If there is a suspension in the rear, would the vibration from the floor still transmit to the kid feet and body?

   Anything about the feet is irrelevant for suspension. But it's also irrelevant because in a balance bike the feet are not traversing the ground in the way the tyres are. At slow speed you're just walking as usual, which clearly doesn't require suspension, but at faster pace the feet are spending most of the time both in the air, only occasionally kicking off the ground. Which means on a balance bike more of the weight is on the saddle than on a pedal bike. Specifically, you just don't have the option to get off the saddle and absorb bumps through the legs on the pedals. And that is actually an argument why rear suspension is somewhat more desirable on a balance bike.
   Whether it's enough of an argument to justify putting suspension on a simple balance bike for small kids is another matter.

3. When comparing to mountain bikes, the suspension is put in different places than the above.

   Yes, but this suspension style is actually the default in motorbikes. It certainly should be considered a proper rear suspension.

   Why don't mountain bikes use the above configuration too?

   It requires a large gap between the bottom bracket and rear wheel. In other words, long chainstays. Mountainbikes generally try to keep the chainstays fairly short for maneuverability / playfulness.
   This suspension design could work well on a cargo / touring bike though.

Rear suspension does help comfort and grip, but on bikes it’s mainly about keeping the wheel planted, comfort is kind of a bonus. On a balance bike, yeah… most bumps still go straight through the kid’s feet since they’re on the ground, so rear suspension only helps a bit (mostly through the seat). That design is basically closer to a seat suspension, not a true full-suspension frame. Mountain bikes don’t use it because it’s less efficient and doesn’t control the wheel as well as proper frame suspension.