Great questions, all of them! No, there need not be any "master file" - but there can be. ;) Let me lay it out for you.

There are two possible models you can build with Syncthing:

• a star-shaped client-server structure places the server at the centre of the network. The server is the central authority that always holds the "master file", i.e. the newest version of the file, and the clients receive it from there (or don't, if they're already up to date). This structure, mind you, does not require dedicated server hardware: if you, say, have a phone, a notebook and a desktop PC, you can also make your phone the "server", i.e. the only device that is connected to all the others and that "serves" everyone the newest version of your password DB.
• a meshed peer-to-peer structure has no centre - just a number of clients that are connected to all other clients. In this network, whichever device has the "newest" version of the file is the device that distributes this newest copy to all connected peers.

Which model you go for depends on how many devices you're planning to hook up. If you only need to keep two devices in sync (e.g. your phone and your notebook), the client-server structure would probably be overkill, and actually Syncthing is typically deployed in serverless peer-to-peer scenarios.

However, if you have a lot of devices to sync, it may be more efficient to go with the client-server setup. Example time to make this more transparent!

Let's say you go with the p2p structure because you only have a phone and a notebook hooked up to Syncthing. You configure:

• on the phone: share a folder with the notebook!
• on the notebook: share a folder with the phone!

But now you want to add another device, e.g. a desktop PC. Now you need to

• add the PC to the phone, and add the share
• add the PC to the notebook, and add the share
• add the phone and the notebook to the PC, and add the share

Otherwise you end up with a partial mesh, i.e. not every device immediately writes its changes to all others.

If you now add a fourth device that also needs to receive the share, you already have to make changes on four devices. And so on and so on. The more devices you add, the more work you need to put in.

TL;DR:

Syncthing can work without a "master file". It can do peer-to-peer shares and work without a server. However, the more devices you need to share to, the more efficient a star-shaped / server-like architecture will become. This, however, does not require dedicated server hardware; you just decide that one of the devices will fulfil the role of the server by hooking all client devices up to it.

One last, minor correction: Syncthing only lets you share folders, not files. So you'd create a "KeePass" folder, toss your DB in there and share the folder to the receiving devices.