If a split test is done on a page with Google Experiments and a visitor returns to make a purchase 3 days later through an email, will the variant that the user first saw get credit for the transaction?
Yes. While Experiments reporting is session based, a user is kept into the test, seeing the same variation the whole time, so when they convert, they will be attributed to the test variation they have been seeing. This is limited to the same browser on the same device, so if the user changes devices (20% of the time for average eCommerce sites by my estimation) or clears cookies (1%) or changes browsers on the the same device (?), than Experiments (and nothing else legal) can track them.
Hope this helps.
PS(I suggest you look at other tools, as Experiments is the last tool I would recommend right now).