Congratulations and welcome to the “can’t own just one “ Ballard club! LoL
To answer some of your questions, yes it appears to have been lined but you should check the twist rate to be sure.  The block will have the serial number at the rear top of the tail. The block does not have the firing pin screw near the front top of the block, which #3s do not have. Most #3s will have a concave top on the top front of the receiver which yours does not have.  The center fires will have the screw. To remove the block take out the lever screw and it should drop down and out. 
It does have the plunger type .22 extractor common to #3s. I would suspect the block is not serial numbered to the rest of the rifle. 
Getting the Ballard to lock up tight with a snap can be difficult and some trial and error. Starting with replacing the lever screw as it may be worn out of round. Then changing the length of the lever link to the block. I’ve also seen where shims have been inserted at the top back of the frame where the block comes up tight. 
Mostly it will be a process of trial and error! And buy a copy of John Dutchers book on Ballards, it’s the Bible on Ballards.

New lever and link screws will often tighten up a Ballard. If not a longer link may work. They are worth restoring to shooting condition as they are fun and satisfying to shoot.

The pics are bigger than the gun. LOL. 
Thanks for all replies. I may have a source to either repair or fabricate a new extractor.
Any source for replacement screws or links? I see some screws on Ebay but think they are old parts.
Also, the failure of the hammer to stay in the half cock position when the action is closed, is there possible damage to the trigger or sear on the hammer? The hammer simply lowers back down to the fired position when the breech is closed and rests fully on the firing pin.

John

What Marlinguy said is good advice.  I might add that if not comfortable with welding up the extractor, give Peter Nagel at Restorative welding a call.  815-732-1249  I and others here have used him and he does good work.

To add to the hammer not catching the half cock, definitely go through what Marlinguy says.  You did not say if cocking manually the hammer catches and holds?  So I will add that If it does, then look at the link and hammer to be sure the link is pushing the hammer back far enough to catch the notch.  Here's a picture of the inside of a block.  Arrow is the area I'm referring too. The picture is of a DST block but the area of the link and hammer should be the same.
Bob

Here's some pics of the inner workings
It looks like there is no half cock notch, or that it is worn down or broken.

Thanks for the input marlinguy. I think some TIG work is in the future. I'm up for some file work.

John

Is that a crack in the frame from top to bottom and through the lever screw?? 
Be careful with that one!
Tom Klinger