Japanese Cotton Sponge Cake - How to stop collapsing?
Try using the actual recipe without substitutions or additions.
I've not made this type of cake, so I cannot answer with 100% certainty.  However, in general I would recommend not tinkering with ingredients the first time you make a recipe, especially when baking.  There are structural reasons for a lot of baking ingredients, and messing with them is a dangerous game.
In particular, fat content and raising agents are pretty important for cake structure, and it sounds like you've altered those right off the bat before making the cake once with its actual recipe.
Half and half has at least 3 times the fat of milk.  This could be having an impact on the cake's ability to rise.
I suspect the main culprit is the baking soda, though.  You've added baking soda and acid to a recipe that has neither, which can cause bubbles to expand and pop rather just staying put, which in turn causes cakes to collapse soon after they come out of the oven. (see this article for details)
The need for cream of tarter when whipping egg whites can be overstated.  Its main functions are to prevent overbeating and stop the foam from collapsing if it is sitting around after being beaten. It is pretty difficult to overbeat egg whites if you are whisking by hand.  If you're using an electric mixer, just keep an eye on it. Because you're only beating to medium-stiff peaks in this recipe and are then using the whites right away, the cream of tarter can be left out of the egg-beating process.
In conclusion, try the recipe as it is written.  Definitely remove the added baking soda and cream of tarter.  If you really have no access to milk, water is a better substitution when cake-baking than half-and-half is. If you really want some dairy in there, I might recommend 25 ml half-and-half + 55 ml water as a better substitution for 80 ml of milk than all half-and-half (this will have a much closer fat content to 80 ml of milk than 90 ml of half-and-half will).
Side note: if you ever do want to add cream of tarter to egg white to help with the beating, you do not have to then add baking soda or another base to balance out this tiny addition of acid.  Leave baking soda out of recipes that don't call for it.