Freewheel and quick release
Definitely a freewheel, and a brand new one too.  I can only see the 6 sided axle nut, and no axle locknut.  

Some designs have the cone nut buried deep inside the freewheel and then a spacer, and finally another nut on the outside, something like this:



Is it possible you replaced a 6 speed freewheel with a 7 speed freewheel, which is one cog wider?  This may require rebalancing the spacers on the axle.  You might be able to move a spacer from the left to the right side, and still have the required amount of axle body poking out.  

Do note that a QR axle should end before it exits the frame - the QR's nut should be clamping on the frame not on the end of the axle.   If in doubt, get it checked over in person.