Chain skipping with new chain and freewheel
You don't say what derailleur and shifter you have, but if they are an indexed Campagnolo system the spacing they're designed for is different in the middle of the cassette to the ends. I had exactly these symptoms with a 10-speed Campagnolo cassette on a SRAM bike.

You will have an array of problems if the mech and shifter's idea of cable pull do not match, particularly with an indexed system.

You will have exactly this problem trying to use any non-7-speed shifter/mech with a 7-speed system.

You can also get these symptoms if the B screw of the derailleur doesn't leave enough clearance between the top jockey wheel and the sprocket on the cassette, but it is more usual for this to affect the easy gears.