Are you saving the "registered" cookie? If a user is registered, but accessing the site on another browser, there would be no cookie and he/she wouldn't know how to login. What language/framework are you using?
Yes and No. Rememberme cookie autologs in when you goto company.sprint.im (if you have logged in before). It however does not log you in directly if you try to goto sprint.im. This is something I probably should work on. A simple login link would solve all the problems.
Yes, you definitely need to create a login/signup links. Just ask yourself this question: "what do I gain and loose by not having a login link?" Not all people read and not all people look at icons, so you usually have to have both icon and text to get all your users to take actions. If you make it less of a guess work for your users, they will appreciate you for it.
I use codeigniter.