The 2" of extruded polystyrene is not too different from the same material used under airport runways. Don't worry that your kids will be bouncing off the ceiling. Two alternative systems without insulation areDri-Core and Delta FL; both types have competitors with similar systems. The concrete screws are not "dug" into the concrete. They're drilled in. The depth of embedment varies with the diameter of the screws you're using. Follow the instructions on the package.

rubber is a good idea, and yes it can go right over concrete. My suggestion is look into a product commonly used in gyms, and indoor sports centers called MONDO. Here is a link that might be useful: Mondo flooring

All LVP is actually waterproof. It's a very solid option for basement flooring because of this. That being said though, remember that you should never leave water sitting on your flooring, whatever that flooring, for weeks on end. The water won't damage you LVP, but it will eventually seep through to the bottom floor and gradually cause damage and become a huge insurance problem. You can measure against this though with the installation of a moisture trap depending on your situation. Another thing about LVP, is the option of ER or Embossed in Register. With ER, the texture of the plank matches up perfectly with the register or the original graphic. This means that every grain is captured in detail, including wood knots. An advantage of this is increased surface area so it protects against slipperiness and you from falling. Here's the main link I used to learn about ER and LVP if you'd like more information. I hope this helps!: https://www.reallycheapfloors.com/blog/the-best-flooring-for-basements/

Since you have chosen warmboard over gypcrete, what was your reasoning behind it besides responsiveness? I am sorry for the confusion. I didn't build a house with Warmboard, I had it in a house where original owner had installed it. It wasn't a house I built. I used it in the sense that I heated the house with it. I looked at the Gypcrete system and somehow feels uneasy about it. Doing a concrete pour over first and second floor does not sound structurally sound to me. Gypcrete has positives and negatives but shouldn't be overlooked because of structurally sound. I think a competent and knowledgeable designer is much more important than which type of system you use. So if you don't have a lot of experienced local people, you have to find experienced people who can help design at a distance. I would start with Radiant Engineering (talk to many companies though). A long somewhat unbiased discussion of radiant: First, it should be noted that radiant heat isn't an efficiency discussion. As people have pointed out many times hydronic radiant systems rarely have a positive payback. They are simply too expensive to install and they never payback if you are installing central air. However, every decision in your house isn't about payback. There are many decisions, such as paint color, cabinet color, cabinet design, many fixtures, etc., that are about your enjoyment of your home. No one has a pool installed because they want it to add value to their house. Radiant fits soundly into that category. Next: A discussion of trade-offs between the different systems. It is important to note that ALL radiant floor systems will provide a comfortable environment to a home that other heating methods struggle to achieve. There are basically five considerations in radiant heat (1) installation costs, (2) operating costs, (3) responsiveness (4) allowable customization and (5) noise (as hot water passes through cold aluminum plates it makes noise). If we look at the different systems we can typically see how they handle each of these things. Warmboard is (1) very expensive to install, (2) it has a low operating cost as it is a warm water system rather than a hot water system, (3) incredibly responsive for radiant heat (able to heat a structure in hours rather than days), (4) low customization and (5) low noise. Underfloor transfer plate systems are (1) moderately expensive to install, (2) low operating costs as they are also warm rather than hot water systems, (3) are somewhat responsive (4) highly customizable and (5) have more expansion noise. Gypcrete overpour is (1) moderately expensive (still cheaper than the two above), (2) have low costs as,they too are warm water systems, (3) not responsive at all, (4) are highly customizable and (5) eliminate almost all noise. Staple up systems are (1) pretty cheap (2) high operating costs as they are a hot water system, (3) not responsive at all, (4) limited customization (they are more limited by physics than design), and (5) are pretty noisy. So looking at Warmboard vs. similar systems - the only real concern from using any aluminum covered plywood system is the noise. Warmboard eliminates expansion noise because of the "special" glue they use to adhere the aluminum to the wood. I can't really tell you how special their glue is, I feel confident they will say super special and their competition will disagree. Warmboard type systems vs. underfloor transfer plate systems - Warmboard is quieter but the loops are not really customizable. Loops are 8" apart with Warmboard, so you can choose not to put warmboard in places but you can't get the loops closer together. With transfer plate systems you can customize the heat of a room by simply moving the loops closer together. This means that you can make the floor right outside the shower warmer than the floor beside the toilet (not many people spend a lot of time standing beside their toilet), but there is more expansion noise. Warmboard is a bit more responsive than transfer plates. Warmboard vs, gypcrete overpour - Gypcrete overpour gives you almost unlimited customization without the noise associated with aluminum transfer plate systems. However, it is not at all responsive. Staple up systems - They are more expensive to operate than any of the above systems but are cheaper to install and easier to maintain. You are not likely to nail through a staple up system with a nice thick transfer plate. Also easier to remodel with staple up systems. TL:DR In the end, if I was ranking a radiant system today - I would probably rank it - underfloor transfer plates > gypcrete > Warmboard. But it is a three way photo finish so it wouldn't take a lot to change the order. Also noisy houses don't bother me at all.