If the other springs are pointed blue steel then this one is a replacement. It's extremely unlikely that the tip has broken off, since the stress there is near zero and the stress at the post is highest. That's why when springs do break, they break at the post. What's probably happened is that the spring was insufficiently seated and has worked its way out of the post as you described. The end of the spring will have been flattened and widened before installation, to give it a press fit into the post. The thing to do is to press it back into place as 1saxman states. Also completely correct that you don't want to hit it with a hammer. The bigger risk than knocking the post off altogether is that you'll move it and then the keywork will bind. The best way to press the spring back into place is with a pair of those slotted pliers he mentions. Unless you have good mechanical skills I wouldn't recommend alternative kludge methods like I used till I got a pair of the slotted pliers.
You'll need to do this chop chop so the spring doesn't wallow out the hole in the post.
As to the trill key, it has zero effect on the springs, it's not even sprung, it just sits there without any spring action on it at all. It is true the G# key touch spring operates in opposition to the spring that lifts the G# pad. That's the way it's designed, every saxophone since the very early 1900s has been made this way. It's called the "articulated G#" and it is what permits you to close the G# pad by closing any RH key while leaving the keytouch depressed. (Unfortunately it's also what allows the G# pad to stick.)
You can take off the trill key if you want but it won't affect anything about the behavior of the horn except that you'll lose that alternative fingering that no one uses anyway. You just need to re-seat, or get re-seated, that spring.