Yes, you will have to submit an online application, and the visa will be issued to you on the same platform as well.

Everyone who travels to Sri Lanka, regardless of their age, require an ETA. If your children hold a separate passport, then their application would be separate from yours, but either way, the applicant must fill in their details as they appear on the passport.

This depends. If your flight has a stopover in Sri Lanka, and you do not plan to exit the transit area of the airport, a transit ETA is not required. However, if you are transiting on your way to a third country by a different mode of transport (eg. arrive by flight and depart by boat) or through a different airport (eg. changing airports whilst still in Sri Lanka), you would need an ETA.

Yes, you can apply for an extension of a visa whilst in Sri Lanka itself. This will be valid for 90 days and can be extended a second time for 90 more.

You can apply for an ETA if you are planning on leaving the airport during your time in transit.

No, you do not - however, you will need to show both of these to the Immigration Officer upon arrival into Sri Lanka.