This mechanism only works if people know which key I have signed. Can I assume that such signing information is stored on key servers?
Yes, certifications are stored on key servers.
Does this mean that I have to resend my key to a key server after having signed someone's key?
You have to upload a key, but not yours, instead upload the certified key (or ship it to the owner, who can decide whether to upload it or not).
You have to differentiate between different kinds of trust.
Certifications (Signatory Trust)
OpenPGP expects trust in identity to be public, thus certifications being shared. "I believe that Alice is really who she claims to be" is a statement based on facts that can be shared without further implications (maybe apart from network analysis and clustering being possible).
On the other, you have to decide whom to trust. Certifications issued are only valid, if a key itself is valid (so you can build a trust path to the other user) and you put trust in the certifications of all the users in-between (or partial trust on different paths). "There's Trust and Then There's Trust and Then There's Trust (a nice article on trust in OpenPGP) called this "trust to vouch".
Sharing this kind of trust might have heavy implications. You don't trust in the capabilities to correctly certify others of your boss, wife/husband? They might get this very wrong. It also reveals social connections at a much deeper level, as you have to know the other person to some extend, while you can safely certify a completely stranger's identity by verifying his identity card/passport.