Arduino Controlled adjustable current 0-1A,18V
You began with the constant current power supply design that Dave shows in the linked video at 20:00. His design is good, as usual. (To save an op-amp, watch later at 28:45 in the video. Alternatively, the buffer and adder could be combined.) You put the solenoid at the input to the LT3080 which will allow its current to be controlled, and this current will be drawn from an adapter that can deliver 18V 1A max.

Whenever using an integrated circuit you should read its data sheet, in this case here http://pdf1.alldatasheet.com/datasheet-pdf/view/349922/LINER/LT3080.html . I think there will always be a current into the Vcontrol pin so unless you tolerate that as minimum current in the solenoid, feed that pin separately direct from the 18V supply.

You may have loaded the Arduino output with too low value resistors. To reduce the 0-5V average to 0-1V, replace the 10kohm resistor connected to the Arduino by 40kohm followed by 10kohm to ground.

If the buffer supplies 1V (check it) and so does the LT3080 to the 1 ohm resistor, the current is 1A as planned.