So I especially tested this on my non-Sims PC and the results were disappointing. It turns out that you can't download items from the Store if Sims 3 is not installed. It simply requires the Launcher, which can't exist without Sims 3 installed. On my laptop (where Sims 3 is installed), when I click 'download' it is showing me a pop-up for me to allow opening the Launcher. The PC (where Sims 3 is not installed) isn't showing me anything, even if I wait for absurdly long time (well, my internet is slow, but not that slow). Answer is, that you can't download Sims 3 Store items to a computer which doesn't have Sims 3 installed, unfortunately.