I believe that any time you pour a carbonated beverage between containers, you lose some of the carbonation because CO2 escapes. Pouring a carbonated beverage through a regular sieve causes CO2 to escape even more, and pouring it through a mesh sieve (at least it looks like a mesh sieve in the picture) will cause even more loss of carbonation.
So I would say yes, you are losing carbonation by using that sieve. If it's to filter out the fruits and what ever is in the kombucha, I would just do my best to have larger chunks that would fit through a regular wide holed sieve.
Yes you definitely can, I have done this with fruit flavored F2 that I thought was done, but upon opening I learned that it wasn't. Since it was still pretty sweet I didn't add any more sugar, I just left it out for about another 2 days, and then opened it and it was more carbonated than it was the first time.
I would think that if you didn't add more sugar and there wasn't that much sugar in it anymore like there was in my case, then it would become more acidic. But as long as there is either already sugar still in it, or you add more sugar, I don't think it will become more acidic than if you would have just left it out without putting it in the fridge in between.
