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.
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.)
* 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.
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.