I can't answer all of your questions, but I will take a stab.

Where are you originating from? That would help us suggest other airlines/options.

You could always just call EVA and they should be able to accommodate your need for more than 4 tickets. You may want to see if they have special group discounts. I think the group size typically has to be larger and since you have little kids they might have lowre child fares anyway. Whether you buy them all at once or not would have no bearing on which seats you end up with. Again a phone call should get you seat assignments however you'd like. Worst case you iron it out at the airport but I would try to do it before that.

The MRT is awesome and cheap. Goes to most of the places you want to go. Rush hour can be a bit crowded so I would try to avoid traveling at those times.

Getting to and from the airport may be a challenge unless you have some one picking you up. You might want to look for car services that can accommodate a group of your size with a van.

As for what to do... There is a zoo. I didn't go but I hear it is pretty good as zoos go. Since I don't have any kids (let alone toddlers) it is hard for me to really make many suggestions. If they are that young they probably won't remember much anyway.