Cannot get chain link out
I'd get a new chain and use a chain tool next time. 

The links have rivets in them which are hard to push out without a chain tool (and other tools can weaken the chain leading to failure, especially with a cockamamie way like you're trying). Given that a cheap chain tool is 10-15 dollars it's a worthy investment relative to the cost of a replacement chain. 

In the second picture, the second vertical link from the top is a quick link. You push the plates together (perpendicular to the table) and then push them in opposite directions (parallel to the table in the direction of the chain). 

You will need a chain tool anyway when you get your new chain in order to size it appropriately; this link will give you appropriate directions. And if you have to install a special rivet (e.g. Shimano) to close the chain, you'll need a tool (*). 

(*) On lower speed chains / old days, people used to push the rivet out partially then re push it back in to close the chain. It is now recommended to use the special closing rivet or a quick link, depending on what the chain manufacturer suggests.