How can I store my connection passwords in putty?
You can't, it's by design.
A.2.8 Does PuTTY have the ability to remember my password so I don't have to type it every time?
No, it doesn't.
Remembering your password is a bad plan for obvious security reasons: anyone who gains access to your machine while you're away from your desk can find out the remembered password, and use it, abuse it or change it.
In addition, it's not even possible for PuTTY to automatically send your password in a Telnet session, because Telnet doesn't give the client software any indication of which part of the login process is the password prompt. PuTTY would have to guess, by looking for words like ‘password’ in the session data; and if your login program is written in something other than English, this won't work.
In SSH, remembering your password would be possible in theory, but there doesn't seem to be much point since SSH supports public key authentication, which is more flexible and more secure. See chapter 8 in the documentation for a full discussion of public key authentication.
You can use Kitty, which is a fork of Putty.
I had this problem a while back, you can use mRemote as an SSH (as well as VNC, RDP and Citrix) client that does.
You can also try Tunnelier which supports saving passwords in profiles. It also functions as an SFTP client.
This is a very bad idea.
It is impossible with Putty alone, as it spawns an actual Window. HOWEVER, if you feel that this is good idea for you for whatever reason, download the port of the ssh Linux client to Windows and pipe in the username/password.
Again that is a bad idea.
A GOOD IDEA would be to setup a set of keys, and this would allow for passwordless login.
You can use Putty connection manager which can be configured using a logon macro to login to servers automatically.