Working off of your three bullet points, I'll address KeePass.
a multiplatform program / browser extension, that works just as seamlessly as LastPass
Multiplatform: the official KeePass 2.25 support Windows XP through 8 plus Mono (i.e. Linux, FreeBSD, MacOS). There are a variety of ports, including KeePassDroid for Android, some for iPhone/iPad, Blackberry, MacOS, J2ME, Linux native, etc.
Just as seamlesly: Not as much (unless the browser extension does this for you), though if you master the "URL" field you can do things like simply switching to the KeePass window and let AutoType and/or Open do it for you - as a non-browser Windows example, you can hit Ctrl-U to open up a Remote Desktop window to the target, switch back to the KeePass window and press Ctrl-V to AutoType username and password; then you're in.
said extension would work in a "decentralized" way one solution I can think of is that it would let me select a file where the passwords are saved and encrypted with a "master" password, and I would share the file online via, say, DropBox
KeePass definitely lets you choose a file for each database, and you can share it via DropBox. KeePass 2.x also natively supports databases via FTP, HTTP and WebDAV, and has a plugin for SCP, SFTP and FTPS. Multiple machines can have the database open for write at once - it asks to synchronize when it detects another alteration as you're saving your own alteration.
KeePass also allows multi-factor authentication (i.e. passphrase + a key file or hardware token), which I would really recommend if you're making your database available to other parties - that way if you accidentally type in your KeePass database password into your DropBox login, someone at DropBox won't have everything they need to open your file. I say this because you mention you're worried about LastPass's integrity.
being 100% open source would be nice
KeePass is GPL licensed.
P.S. if you do use KeePass, upon creating each database, please go into File, Database Settings, Security, Key transformation rounds, and change that to a large number (start by pressing the "1 second" button/link and then multiply whatever that is by 2 to 12 or more; even on a decent smartphone ten million rounds doesn't take but a handful of seconds, and it'll slow an attacker down quite a bit.
P.P.S. you can also change a global setting on the main Windows KeePass program under Tools, Options, Security, Enter Master Key on Secure Desktop.