No, they don't sync yet, but they will in a future version. Here's the plan:
See the next two questions on how to move data between the online and offline versions.
Paying for the new desktop version gives you a number of months free for the online one.
The number of months you get is (price you paid / 3.75)
This is a manual process for now, so get in touch with me to receive your free months.
The online version currently works on all devices. Just visit https://gingkoapp.com/login from your device.
When I release v2 of the online version, it'll work the same way.
A native mobile version would be then next step, but that's still a long ways off.
The free trial is exactly the same as the full version, except that the free trial expires once you've opened Gingko on 30 different days.