Life and Lineage uses a per-user database approach for its remote storage, this provides better separation of users and therefore privacy, and it makes requesting the removal of your data simpler, you can even request the database dump if you so choose. This isn't particularly expensive, but not free for n users, so use of this feature requires a purchase of an IAP(in-app purchase) - this can be the specific IAP for the remote save feature, and any other IAP will also unlock this feature.

You can use the above mentioned remote-backups to save progress between devices/platforms.

Currently, the only time you need to be online is for remote save access. There are plans for pvp, which will require an internet connection, but this is not implemented at time of writing.

Microtransactions are not required to play or complete the game, the game can be fully completed without spending any money, however 2 of the classes(necromancer and ranger) are pay-walled. Microtransactions are supported cross-platform, so no need to pay for each device, you simply need to login to your gmail/apple/email account. This would require first creating a character, signing in under options>remote backups first.