* The database in encrypted with your master password.
* You can optionally also encrypt it with static "Key File" that are on all your devices but not in Dropbox.
B) You choose which 3rd Party to trust. There are many options with different security/trust/threat models.
(Example: lately I've been using an encrypted share in Resilio Sync where the "cloud" option for me is a dumb VPS that can share the folder torrent but does not have decryption keys into the contents.)
On desktops/laptops, if you're pointing KeePass at a database file in a Dropbox-synced folder, then it's automatic.
I ran into this when checking unexpected files on a client's system.
I've been looking for a alternative with somewhat parity with lastpass with a better security policy.
Cons: NOT open-source, paid cellphone apps.
So a slew is not necessarily accurate. But I take you're point, it's definitely a similar weak point that LastPass has.
Put your keyfile on Dropbox/OneDrive/whatever so it syncs to all your computers.
Keepass2Android works great and can read from most cloud storage solutions.
Don't know about iPhone.
Edit: It also has a lot of neat plugins. I use one for storing ssl certificates, which also supports key forwarding to putty.