Can someone inform me on how to achieve this type of patina on a furniture project I am working on

You took the wind out of my sails with that last paragraph. I was going to suggest using a nut/bolt.

As far as it causing problems when the tabletop is lowered, you can recess the bolt head. Yeah, you could make the cavity in which the bolt will stay on the underside of the tabletop, but maybe you'll drill too far or crack/scratch the tabletop if you chisel it out. It could end badly... which is why you should instead house the bolt head on the top-side of the cross member.

Brown = table top
Black = underlying cross member

The one on the left is what you were talking about. The one on the right is what you want to do.

Also, use a washer under the bolt head, etc.

It depends on whether the stickiness was caused by the ball leaving a residue on the finish or the ingredients in the ball (probably solvents) ate into the finish.

If the former, you may be able to remove a residue using a wood cleaner like Murphy's Oil Soap (just one example). These cleaners are generally safe for wood finishes. If this fails, you could try stronger cleaners like Goo Gone. This type of cleaner is more likely to adversely affect finish, so try it in an inconspicuous place. However, if the milder soap doesn't clean it, you probably are risking the finish anyway.

If that doesn't work, you can try solvents, beginning with mineral spirits, but you are highly at risk of damaging finish. After wiping with the solvent, carefully wash off, again using wood soap or a mild detergent.

You can also try alcohol and lacquer thinner, either of which is highly likely to eat away the finish, but by then you are past preservation.